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\begin{center}
\vskip 1cm{\LARGE\bf 
Generalized $j$-Factorial Functions, Polynomials, and Applications \\ 
}
\vskip 1cm
\large
Maxie D. Schmidt \\ 
University of Illinois, Urbana-Champaign\\
Urbana, IL  61801\\
USA\\
\href{mailto:maxieds@gmail.com}{\texttt{maxieds@gmail.com}}
\end{center}

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\begin{abstract} 
The paper generalizes the traditional single factorial function to 
integer-valued multiple factorial ($j$-factorial) forms. The 
generalized factorial functions are defined recursively as triangles of 
coefficients corresponding to the polynomial expansions of a subset of 
degenerate falling factorial functions. The 
resulting coefficient triangles are similar to the classical sets of 
Stirling numbers and satisfy many analogous finite-difference 
and enumerative properties as the well-known combinatorial triangles. The 
generalized triangles are also 
considered in terms of their relation to elementary symmetric polynomials and 
the resulting symmetric polynomial index transformations. 
The definition of the Stirling convolution polynomial sequence is 
generalized in order to enumerate the 
parametrized sets of $j$-factorial polynomials and 
to derive extended properties of the $j$-factorial function expansions. 

The generalized $j$-factorial polynomial sequences considered lead to 
applications expressing key forms of the $j$-factorial functions in 
terms of arbitrary partitions of the $j$-factorial function expansion triangle 
indices, including several identities related to the 
polynomial expansions of binomial coefficients. Additional 
applications include the formulation of closed-form identities and 
generating functions for the Stirling numbers of the first kind and 
$r$-order harmonic number sequences, 
as well as an extension of Stirling's approximation for the 
single factorial function to approximate the more general 
$j$-factorial function forms. 
\end{abstract} 

\section{Notational Conventions} 
\label{subSection_Intro_notations} 

\nocite{KNUTHNOTATION} 
Donald E.\ Knuth's article \papertitle{Two Notes on Notation} 
\cite{KNUTHNOTATION} establishes several of the 
forms of standardized notation employed in the article. 
In particular Knuth's notation for the 
\keyword{Stirling number} triangles and the 
\keyword{Stirling polynomial} sequences \cite{GKP,CVLPOLYS} are 
used to denote these forms and reasonable extensions of these conventions 
are used to denote the generalizations of the forms established by this 
article. 
%% \cite[p.\ 18]{KNUTHNOTATION}: 
%% \keyword{q--generalization of the Stirling triangle coefficients} 
The usage of notation for standard mathematical functions is explained 
inline in the text where the context of the relevant forms apply 
\cite{NISTHB}. 
The following is a list of the other main notational conventions employed 
throughout the article. 
     \begin{enumerate} 
          \newcommand{\itemlabel}[1]{\textbf{\underline{#1}:}} 
          \renewcommand{\labelenumi}{$\bullet$} 
          \setlength{\itemsep}{-1mm} 
          \item \itemlabel{Indexing Sets} The \keyword{natural numbers} are 
                denoted by the set notation $\Nset$ and are equivalent 
                to the set of non-negative integers, where the 
                set of integers is denoted by the similar 
                blackboard set notation for $\Zset$. The 
                standard set notation for the 
                real numbers ($\mathbb{R}$) and 
                complex numbers ($\mathbb{C}$) is used as well to 
                denote scalar and approximate constant values. 
          \item \itemlabel{Natural Logarithm Functions} The 
                \keyword{natural logarithm} function is denoted $\Log(z)$ 
                in place of $\ln(z)$ in the series expansion properties 
                involving the function. Similarly $\Log(z)^{k}$ denotes 
                the natural logarithm function raised to the 
                $k^{th}$ power. 
          \item \itemlabel{Iverson's Convention} The notation 
                $\Iverson{\mathtt{condition}}$ for a 
                boolean-valued input condition 
                represents the value $1$ (or $0$) where the input condition 
                evaluates to \texttt{\textbf{True}} 
                (or \texttt{\textbf{False}}). Iverson's convention is 
                used extensively in the \emph{Concrete Mathematics} 
                reference and is a comparable replacement for 
                \keyword{Kronecker's delta} function for multiple 
                pairs of arguments. For example, the notation 
                $\Iverson{n = k}$ is equivalent to $\KDelta{n}{k}$ and 
                the notation $\Iverson{n = k = 0}$ is equivalent to 
                $\KDelta{n}{0}\KDelta{k}{0}$. 
          \item \itemlabel{Sequence Enumeration and Coefficient Extraction} 
                The notation \\ 
                $\langle g_n \rangle \mapsto \{g_0,\ g_1,\ g_2,\ \ldots\}$ 
                denotes a sequence indexed over the natural numbers. 
                Given the generating function $F(z)$ representing the formal 
                power series (also generating series expansion) that 
                enumerates $\langle f_n \rangle$, the 
                notation $[z^n] F(z) := f_n$ denotes the 
                series coefficients indexed by $n \in \Nset$. 
          \item \itemlabel{Fixed Parameter Variables} For an indexing 
                variable $n$, the notation $\Nc$ 
                is employed to represent a 
                fixed parameter in a formula or generating function that 
                is treated as a constant and that 
                is only assigned the explicit value of the respective 
                non-constant indexing variable after all 
                other variables and indices have been input and processed 
                symbolically in a relevant form. In particular the fixed $\Nc$ 
                variable should be treated as a constant parameter in series 
                or generating function closed-forms, even when the 
                non-constant form of $n$ refers to a particular coefficient 
                index in the series expansion. The footnote %\refft{footnote_Nc_Mc_notation} 
                on p.\ \pageref{footnote_Nc_Mc_notation} clarifies the 
                context and particular utility of the fixed 
                parameter usage in a specific example inline in the text. 
     \end{enumerate} 

\section{Introduction}
\label{Section_intro}

The parametrized \keyword{multifactorial} (\keyword{$j$-factorial}) 
functions studied in this article generalize the 
standard classical \keyword{single factorial} 
\citeseqnum{A000142} and \keyword{double factorial} 
\citeseqnumIII{A001147}{A000165}{A006882}
functions and are characterized by the analogous recursive property in 
\eqref{MF_jFact_rdef}. 
\begin{equation} 
\label{MF_jFact_rdef} 
n!_{(j)} := n\ (n-j)!_{(j)} \Iverson{n \geq j}+\Iverson{0 \leq n < j} 
\end{equation} 
The classical \keyword{generalized falling factorial} function, 
$\FFact{z}{\alpha}{n}$, studied extensively by Adelberg and several others 
\cite{ADELBERGA,CK,CS,HS,K,CGSDEGENNUMSEQS}, can be defined analytically in 
terms of the \keyword{gamma function} in \eqref{FFact_analytic_def} and 
by the equivalent product expansion form in \eqref{FFact_product_def}. 
The function can also be expressed by the equivalent 
finite-degree polynomial expansion in $z$ with 
coefficients given in terms of the unsigned triangle of 
\keyword{Stirling numbers of the first kind} 
\citeseqnum{A094638} in equation \eqref{FallingFact_classicial_S1_ident}. 
\begin{align} 
\label{FFact_analytic_def} 
\FFact{z}{\alpha}{n} & := \frac{\alpha^n\ \Gamma(\frac{z}{\alpha})}{ 
     \Gamma(\frac{z}{\alpha}-n)} + \Iverson{n = 0} \\ 
\label{FFact_product_def} 
     & \phantom{:}= \prod_{i=0}^{n-1} (z-i\alpha)+ \Iverson{n = 0} \\ 
\label{FallingFact_classicial_S1_ident} 
     & \phantom{:}= \sum_{k=1}^{n} \gkpSI{n}{k} 
     (-\alpha)^{n-k} z^k + \Iverson{n = 0} 
\end{align} 
The definitions of \eqref{FFact_analytic_def} and \eqref{FFact_product_def} 
extend a well-known simplified case of the 
\keyword{falling factorial} function for $\alpha = 1$, 
commonly denoted by the equivalent forms $x^{\underline{n}} = x! / (x-n)!$ and 
$\Pochhammer{x}{n}$ \cite{GKP,UC}. 
Other related factorial function variants 
include the \keyword{generalized factorials of t of order n and increment h}, 
denoted $t^{(n,\ h)}$, considered by C.\ Charalambides 
\cite[\Section 1]{q-DIFFSq-FACTS}, the forms of the 
\keyword{Roman factorial} and \keyword{Knuth factorial} functions defined by 
Loeb \cite{LOEBBINOM}, and the \keyword{q-shifted factorial} functions 
defined by McIntosh \cite{q-SHIFTEDFACTS} and Charalambides 
\cite[(2.2)]{q-DIFFSq-FACTS}. 

This article explores forms of the polynomial expansions corresponding to 
a subset of the generalized, integer-valued falling factorial functions 
defined by \eqref{FFact_product_def}. The finite-difference forms studied 
effectively generalize the Stirling number identity of 
\eqref{FallingFact_classicial_S1_ident} for the class of 
\keyword{degenerate falling factorial} expansion forms given by 
$\FFact{x-1}{\alpha}{n}$ when $\alpha$ is a positive integer. 
The treatments offered in many standard 
works are satisfied with the analytic gamma function representation 
of the full falling factorial function expansion. In contrast, the 
consideration of the generalized factorial functions considered by 
this article is motivated by the need for the precise definition of 
arbitrary sequences of the coefficients that result from the variations of the 
finite-degree polynomial expansions in $z$ originally defined by equation 
\eqref{FallingFact_classicial_S1_ident}. 

For example, in a motivating 
application of the research it is necessary to extract only the 
even powers of $z$ in the formal polynomial expansion of the 
double factorial function over $z$. 
The approach to these expansions is similar in many respects to 
that of Charalambides' related article where 
the expansion coefficients of the \keyword{generalized $q$-factorial} functions 
are treated separately from the forms of the full factorial function products 
\cite[\Section 3 and \Section 5]{q-DIFFSq-FACTS}. 
The coefficient-based definition of the falling factorial function variants 
allows for a rigorous and more careful study of the individual 
finite-degree expansions that is not possible from the purely analytic 
view of the falling factorial function given in terms of the 
full product expansion and infinite series representations of the 
gamma function [cf.\ \eqref{FFact_analytic_def}]. 

The exploration of the $j$-factorial function expansions begins in 
\sref{subSection_Fa_Fa2_prelim_defs} by motivating the 
recursive definition of the coefficient triangles for the polynomial 
expansions of the (degenerate) factorial functions defined in the form of 
equation \eqref{FallingFact_classicial_S1_ident}. The article then expands 
the properties of the factorial function expansions in terms of 
finite-difference identities and enumerative properties in 
\sref{Section_Fa_rdefs}, relations to 
transformations of \keyword{elementary symmetric functions} in 
\sref{Section_Symm_poly_topics}, and 
in the forms of the \keyword{$j$-factorial polynomials} that generalize the 
sequence of \keyword{Stirling polynomials} in \sref{Section_MF_Fa_polynomials}. 
A number of interesting applications and examples are 
considered as well, with particular emphasis on the forms discussed in 
\sref{subSection_SymmTransform_and_HarmonicNumber_Apps} and 
\sref{Section_JFact_Poly_Applications}. 

\section{Finite Difference Representations for the 
         $j$-Factorial Function Expansion Coefficients} 
\label{Section_Fa_rdefs} 

\subsection{Triangle Definitions} 
\label{subSection_Fa_Fa2_prelim_defs} 

Consider the coefficient triangles indexed over $n, k \in \Nset$ and 
defined recursively by equations \eqref{Fa_rdef} and 
\eqref{Fa2_rdef} \cite[cf.\ (1.2)]{K}. 
\begin{align} 
\label{Fa_rdef} 
\FcfII{\alpha}{n}{k} & = (\alpha n+1-2\alpha)\FcfII{\alpha}{n-1}{k} + 
     \FcfII{\alpha}{n-1}{k-1} + \Iverson{n = k = 0} \\ 
\label{Fa2_rdef} 
\FIIcfII{\alpha}{n}{k} & = (\alpha k+1-\alpha)\FIIcfII{\alpha}{n-1}{k} + 
     \FIIcfII{\alpha}{n-1}{k-1} + \Iverson{n = k = 0} 
\end{align} 
The \quotetext{\keyword{triangular}} recurrences defining the 
\keyword{$\alpha$-factorial triangles} are special cases of 
a more general form in equation \eqref{eqn_gkp_Y_rform_def} 
that includes well-known classical combinatorial sets \cite[Ch.\ 6]{GKP} 
such as the \keyword{Stirling cycle numbers} (first kind) 
\citeseqnumII{A008275}{A094638}, also defined by \eqref{Fa_rdef} 
when $\alpha := 1$, the \keyword{Stirling subset numbers} 
(second kind) \citeseqnum{A008277}, the 
\keyword{Eulerian numbers} for permutation \keyword{\quotetext{ascents}} 
\citeseqnum{A066094}, and the \keyword{\quotetext{second-order} 
Eulerian numbers} \citeseqnum{A008517} 
[cf.\ \sref{subsubSection_ApplTo_EulerianTriangleRowSums}]. 
The unsigned triangles corresponding to a positive integer parameter $\alpha$ 
are unimodal over each row and are strictly increasing at each fixed 
column for sufficiently large row index $n$ \cite[\Section 4.5]{GFOLOGY}. 
The signed coefficient analog of the triangle in \eqref{Fa_rdef} is defined 
recursively as \eqref{FaBar_rdef} and may be expressed in terms of the 
unsigned triangle by the conversion formulas in equation 
\eqref{eqn_Fa_FaBar_conversion_form_ident}. 
\begin{align} 
\label{FaBar_rdef} 
 & \FcfBarII{\alpha}{n}{k} = (2\alpha-\alpha n-1) \FcfBarII{\alpha}{n-1}{k}+
     \FcfBarII{\alpha}{n-1}{k-1}+\Iverson{n = k = 0} \\ 
\label{eqn_Fa_FaBar_conversion_form_ident} 
 & \FcfBarII{\alpha}{n}{k} = (-1)^{n-k} \FcfII{\alpha}{n}{k} \qquad\iff\qquad  
     \FcfII{\alpha}{n}{k} = (-1)^{n-k} \FcfBarII{\alpha}{n}{k} 
\end{align} 

The implicit 
interpretation of the \eqref{Fa_rdef} triangle as expansion coefficient sets 
is demonstrated by considering the motivation for the following procedure. 
Let the polynomial $p_n(x)$ correspond to 
the $n^{th}$ distinct polynomial expansion of the 
\keyword{$\alpha$-factorial} function, $(x-1)!_{(\alpha)}$, and define the 
polynomial coefficients as 
$[x^{k-1}] p_n(x) := \FcfII{\alpha}{n}{k}$. Observe that provided a 
polynomial (row) index $n$, the coefficient forms corresponding to 
subsequent polynomial expansions are formed by the 
multiplication of a linear factor in $x$ with the existing polynomial. 
The next equation defines the triangle of coefficients that result from the 
expansions of this form \cite[cf.\ Thm.\ 3.2]{q-DIFFSq-FACTS}. 
\begin{equation*} 
[x^{k-1}]p_n(x) = (\alpha n+1-2\alpha)\ [x^{k-1}] p_{n-1}(x) + 
     [x^{k-2}] p_{n-1}(x) 
\end{equation*} 
It follows that for rows indexed by $n \in [1,\ \infty) \subseteq \Nset$ and 
columns indexed by $k \in [1,\ \infty) \subseteq \Nset$, the 
polynomial expansions yield an identical recursive definition of the 
$\alpha$-factorial coefficient triangles to that given by 
\eqref{Fa_rdef} [cf.\ \eqref{FallingFact_gen_Fa_ident}]. 

In order to evaluate the factorial function expansions numerically, 
consider that the range of natural numbers that correspond to any 
distinct $\alpha$-factorial polynomial expansion in $s$ is a function of 
$\alpha$: 
there is exactly one $n \in \mathbb{N}$ for each single factorial function 
expansion, two $n \in \mathbb{N}$ for each double factorial function 
expansion, and so on for each positive integer value of $\alpha$. 

\subsection{Finite-Difference Properties} 
\label{subSection_FiniteDiffIdents} 

Many analogs of the classical Stirling number identities and 
related combinatorial properties for the triangles in equations 
\eqref{Fa_rdef} and \eqref{Fa2_rdef} are generalized by the 
analogous forms in the following discussions 
[cf.\ \sref{subSection_GenStirlingNums_forms_and_diffs}]. 
A number of additional identities and forms related to the 
Stirling number forms, including relations to the 
\keyword{Bell numbers} \citeseqnum{A000110}, \keyword{Lah numbers} 
\citeseqnum{A008297}, \keyword{multi-poly Bernoulli numbers}, and 
\keyword{Tanh numbers} \citeseqnum{A111593}, 
are discussed in the references by Agoh and several others 
\cite{GENS2CVLIDENTS,CVLIDENTSBNUMS,STIRLINGNUMREPS,JISRICCIA11PDF} 
\cite[(3.5)]{CS} \cite[Table 202]{GKP} 
\cite[\Section 2 and \Section 3]{MULTIPOLYBNUMS} \cite[\Section 3.1]{HS} 
\cite[Thm.\ 2.1 and (2.4)]{COMBIDSDUALS}. 
The discussions and properties of the 
$\alpha$-factorial polynomials given in 
\sref{Section_MF_Fa_polynomials} also provide a number of identities involving 
\keyword{generalized Bernoulli polynomials} and other functions that may 
be applied to the forms of \eqref{Fa_rdef} and \eqref{Fa2_rdef}. 
The next section in \sref{Section_Symm_poly_topics} contains 
detailed discussions of the \eqref{Fa_rdef} triangle properties as well. 

The initial characteristic finite-difference properties defining the 
Stirling number sets are mirrored by the 
generalized $\alpha$-factorial triangles as given by the 
following equations in \eqref{FallingFact_gen_Fa_ident} and 
\eqref{xn_gen_Fa2_ident} \cite[\Section 1.2.6]{TAOCP}. 
\begin{align} 
\label{FallingFact_gen_Fa_ident} 
\FFact{x-1}{\alpha}{n} & \phantom{:}= 
     \sum_{k=0}^{n-1} \FcfII{\alpha}{n}{k+1} (-1)^{n-1-k} x^{k} \\ 
\label{xn_gen_Fa2_ident} 
x^n & := \sum_{k=0}^n \FIIcfII{\alpha}{n}{k} \FFact{x-1}{\alpha}{k} 
\end{align} 
Depending on the application it may be convenient to define the 
parameter $\alpha$ in equations \eqref{Fa_rdef} and \eqref{Fa2_rdef} over the 
rational numbers. This slight generalization in form will still result in the 
correct form for the interpretation given by the 
product expansion of \eqref{FallingFact_gen_Fa_ident}. 
For example, the book \booktitle{Concrete Mathematics} considers the 
form of $r^{\underline{k}} (r-1/2)^{\underline{k}}$ related to the 
\keyword{central binomial coefficients} \cite[\Section 5.3]{GKP}. 
One additional special case of equation \eqref{Fa_rdef} occurs when 
$\alpha := 0$ where the form provides the recursive definition for 
\keyword{Pascal's triangle}. The case degrades nicely in the 
context a $0$-factorial function where the polynomial factors in the 
expansion remain constant in form over all $n$. The form of 
equation \eqref{FallingFact_gen_Fa_ident} for this case then 
effectively defines the \keyword{binomial theorem} in reverse: 
$\sum_{k} \FcfII{0}{n}{k}\ s^{k-1} = (s+1)^{n-1}$. 

The Stirling number inversion identities \cite[Table 264]{GKP} 
are generalized by the forms of equations 
\eqref{Fa_Fa2_inversion_idents_V1} and \eqref{Fa_Fa2_inversion_idents_V2}. 
\begin{align} 
\label{Fa_Fa2_inversion_idents_V1} 
\sum_{k=0}^n \FcfII{\alpha}{n}{k}\FIIcfII{\alpha}{k}{m} (-1)^{n-k} & 
     = \Iverson{m = n} \\ 
\label{Fa_Fa2_inversion_idents_V2} 
\sum_{k=0}^n \FIIcfII{\alpha}{n}{k}\FcfII{\alpha}{k}{m} (-1)^{n-k} & 
     = \Iverson{m = n} 
\end{align} 
The orthogonality relation for the Stirling numbers is preserved in 
these properties for the generalized triangles and 
defines the analogous result of corollary \eqref{eqn_fg_iff_cor} 
\cite[cf.\ (1.1)]{JISRICCIA11PDF} \cite{GKP}. 
\begin{equation} 
\label{eqn_fg_iff_cor} 
f(n) = \sum_{k=0}^n \FcfII{\alpha}{n}{k} (-1)^k g(k) \quad\iff\quad 
g(n) = \sum_{k=0}^n \FIIcfII{\alpha}{n}{k} (-1)^k f(k) 
\end{equation} 
 
The identities of equations \eqref{CM_6.25_ident_gen} and 
\eqref{CM_6.24_ident_gen} generalize classical Stirling number properties 
as well \cite[Table 265]{GKP}. 
\begin{align} 
\label{CM_6.25_ident_gen} 
\sum_{k=0}^n \FcfII{\alpha}{n+1}{k+1}\FIIcfII{\alpha}{k}{m} (-1)^{m-k} & 
     = \frac{\alpha^{n-m}\ \Gamma\left(n+\frac{1}{\alpha}\right)}{
     \Gamma\left(m+\frac{1}{\alpha}\right)} \Iverson{n \geq k} \\ 
\label{CM_6.24_ident_gen} 
\sum_{k=0}^{n+1} \FIIcfII{\alpha}{n+1}{k+1}\FcfII{\alpha}{k}{m} (-1)^{m-k} & 
     = \sum_{j=0}^{n-k} \binom{k-1+j}{k-1} \alpha^j + 
     \Iverson{k = 0} 
\end{align} 
 
The generalized first triangle in \eqref{Fa_rdef} can be 
expressed in terms of the 
Stirling numbers of the first kind through the following identities. 
\begin{align} 
\label{eqn_Fa_in_terms_of_S1_sum_V1} 
\FcfII{\alpha}{n}{k} & = \sum_{j=0}^{n-k} \gkpSI{n-1}{k-1+j}\ 
     \binom{k-1+j}{k-1}\ \alpha^{n-k-j} +\Iverson{n = k = 1} \\ 
\notag 
\FcfII{\alpha}{x}{x-n} & = \sum_{k=0}^n \gkpSI{x}{x-k}\ 
     \frac{\alpha^k\ \Gamma(x-k)\ (1-\alpha)^{n-k}}{
     \Gamma(x-n)\ \Gamma(n-k+1)} 
\end{align} 
The generalized second triangle in \eqref{Fa2_rdef} can be 
expressed in terms of the \keyword{Stirling numbers of the second kind}
through the following equations \cite[(3.4) and (3.5)]{CVLIDENTSBNUMS}. 
\begin{align} 
\label{eqn_Fa2_in_terms_of_S2_sum} 
\FIIcfII{\alpha}{n}{k} & = \sum_{j=0}^{n-k} \phantom{\sum} 
     \gkpSII{k-1+j}{k-1}\ \binom{n-1}{k-1+j}\ \alpha^j \\ 
\notag 
   & = \sum_{i=0}^{k-1} \sum_{j=0}^{n-k} \binom{n-1}{k-1+j} 
     \binom{k-1}{i}\ \frac{(-1)^i\ \alpha^j\ (k-1-i)^{k-1+j}}{(k-1)!} \\ 
\notag 
   & = \sum_{i=1}^{k-1} \sum_{j=0}^{n-k} \binom{n-1}{k-1+j} 
     \binom{k-2}{i-1}\ \frac{(-1)^{k-1+i}\ \alpha^j\ i^{k-2+j}}{(k-2)!} 
\end{align} 
 
Let the \keyword{linear differential operator}, $\{D^k\}[f(\Nc)]$, be 
defined such that for integer $k \geq 1$ the operator denotes the 
$k^{th}$ partial derivative of $f$ with respect to $\Nc$ and 
for all other $k$, $\{D^k\}[f(\Nc)] := f(\Nc) \Iverson{k = 0}$. 
Additional properties of the generalized first triangle are 
then given in pairs below for the finite $n \rightarrow \Nc$ and 
corresponding finite-difference 
\keyword{\quotetext{binomial derivative}} formulas. 
\begin{align} 
\notag 
\FcfII{\alpha}{n}{k} & = {\left\{\frac{D^{k-1}}{(k-1)!\ \alpha^{k-1}}+
     \frac{D^{k-2}}{(k-2)!\ \alpha^{k-2}}\right\} \left[ 
     \sum_{i=0}^{n-3} (\alpha \Nc+1-2\alpha)^{i+1} (-\alpha)^{n-3-i} 
     \gkpSI{n-2}{i+1}\right]} \\ 
\label{Fa_S1_BinomDeriv_ident} 
    & =\phantom{+}\sum_{i=0}^{n-3}\sum_{r=0}^{i+1} \gkpSI{n-2}{i+1} 
      \binom{i+1}{r}\binom{r}{k-1}\ (-1)^{n-3-i} \alpha^{n+r-2-i-k} \Nc^{r+1-k} 
      (1-2\alpha)^{i+1-r} \\ 
\notag 
    & \phantom{=}+\sum_{i=0}^{n-3}\sum_{r=0}^{i+1} \gkpSI{n-2}{i+1} 
      \binom{i+1}{r}\binom{r}{k-2}\ (-1)^{n-3-i} \alpha^{n+r-1-i-k} \Nc^{r+2-k} 
      (1-2\alpha)^{i+1-r} \\ 
\notag 
\FcfII{\alpha}{n}{k} & = {\left\{\frac{D^{k-1}}{(k-1)!\ \alpha^{k-1}}+
     \frac{D^{k-2}}{(k-2)!\ \alpha^{k-2}}\right\} \left[ 
     \sum_{i=0}^{n-2} (\alpha \Nc+2-2\alpha)^i (-1)^{n-i} 
     \FcfII{\alpha}{n-1}{i+1}\right]} \\ 
\label{Fa_Fa_BinomDeriv_ident} 
   & =\phantom{+}\sum_{i=0}^{n-2}\sum_{r=0}^{i} \FcfII{\alpha}{n-1}{i+1} 
     \binom{i}{r}\binom{r}{k-1}\ (-1)^{n-i} \alpha^{r+1-k} \Nc^{r+1-k} 
      2^{i-r} (1-\alpha)^{i-r} \\ 
\notag 
   & \phantom{=}+\sum_{i=0}^{n-2}\sum_{r=0}^{i} \FcfII{\alpha}{n-1}{i+1} 
     \binom{i}{r}\binom{r}{k-2}\ (-1)^{n-i} \alpha^{r+2-k} \Nc^{r+2-k} 
      2^{i-r} (1-\alpha)^{i-r} 
\end{align} 
The second of the listed \quotetext{binomial derivative} formulas may be 
considered as an \quotetext{\keyword{involution of sorts}} since the 
coefficient form of $\FcfII{\alpha}{n}{k}$ is given in terms of 
coefficients from the same triangle. The identity is also of 
particular interest since the involution-like phrasing
results in the applications for the $\alpha$-factorial polynomials 
discussed in \sref{subsubSection_FirstPolyResults_v1} and 
\sref{subsubsection_BPolyEnums_of_FirstTriangle} that are based on the 
alternate forms of the involution identities derived in 
\sref{subSection_init_Fa_symm_idents}. 

Finally, the sums of the first coefficient triangle rows 
indexed over the integer $n \geq 1$ have the 
generalized forms given by the next pair of equations in 
\eqref{Fa_row_sum_ident} and \eqref{Fa_row_sum_ident_V2} 
\cite[cf.\ (6.9)]{GKP} 
\begin{align} 
\label{Fa_row_sum_ident} 
 & \sum_{k=0}^n \FcfII{\alpha}{n}{k} = {\alpha^{n-1}}\ 
     \frac{\Gamma\left(n-1+\frac{2}{\alpha}\right)}{ 
     \Gamma\left(\frac{2}{\alpha}\right)} \\ 
\label{Fa_row_sum_ident_V2} 
 & \sum_{k=0}^n \FcfII{\alpha}{n}{k} (-1)^{n-k} = 
   \sum_{k=0}^{n} \FcfBarII{\alpha}{n}{k} = 0 
\end{align} 

\subsection{Enumerative Properties} 
\label{subSection_Fa_GF_form_idents} 

The primary motivation for considering the initial enumerations given in this 
section is to generalize the important generating function identities 
for the classical Stirling number triangles summarized in Table $351$ of 
\booktitle{Concrete Mathematics} \cite[\Section 7.4]{GKP}. In general, the 
resulting generalizations for the triangles in \eqref{Fa_rdef} and 
\eqref{Fa2_rdef} are more complex than that of the original 
Stirling number identities, though the 
results are key in characterizing the behavior of the generalized 
$\alpha$-factorial function expansions. Additional enumerative properties 
for the triangles are established in the discussions of 
\sref{Section_Symm_poly_topics}, \sref{Section_MF_Fa_polynomials}, and 
\sref{subSection_MF_expansion_polys}. 

To begin with, consider the next generalization of the classical 
identity of \eqref{eqn_CMathTable351GF_7.48} \cite[(7.48)]{GKP} 
in equation \eqref{eqn_CM_7.48_GF_gen_V2} 
defined over the upper index $m \in [1,\ \infty) \subseteq \Nset$. 
\begin{align} 
\label{eqn_CMathTable351GF_7.48} 
 & \sum_{n \geq 0} \FcfII{\phantom{\alpha}}{m}{n} z^{n\phantom{-1}} = 
     z^{\overline{m}} = z(z+1) \cdots (z+m-1) \\ 
\label{eqn_CM_7.48_GF_gen_V2} 
 & \sum_{n \geq 0} \FcfII{\alpha}{m}{n} z^{n-1} = 
     \frac{\alpha^{m-1} \Gamma(m-1+\frac{z+1}{\alpha})}{ 
     \Gamma(\frac{z+1}{\alpha})} 
%\notag 
% & \sum_{n=0}^{\infty} \FcfII{\alpha}{m}{n} (-1)^{m+n} z^{n-1} = 
%       \frac{(-\alpha)^{m-1} \Gamma(m-1+\frac{(1-z)}{\alpha})}{
%       \Gamma(\frac{(1-z)}{\alpha})} 
\end{align} 
Next, let the function $f_m$ be defined by 
equation \eqref{eqn_fm_CM_7.46_extension_fn}. 
\begin{equation} 
\label{eqn_fm_CM_7.46_extension_fn} 
f_m := \sum_{k=0}^{m} \sum_{j=0}^{k} \sum_{i=0}^{j} \FIIcfII{\alpha}{m}{k} 
     \gkpSI{k}{j} \gkpSII{j+1}{i+1} \frac{(-\alpha)^{k-j}\ i!\ z^i}{(1-z)^i} 
\end{equation} 
The form of the identity in equation \eqref{eqn_CMathTable351GF_7.46} 
\cite[(7.46)]{GKP} can be extended by the forms of 
\eqref{eqn_CM_7.46_gen_ident1} and \eqref{eqn_CM_7.46_gen_ident2}, as 
well as by the form of equation \eqref{eqn_CM_7.46_gen_ident3} for positive 
integer $m$ \cite{ADELBERGA} \cite[cf.\ $A_{k, i}$]{STIRLINGPOLYS}. 
\begin{align} 
\label{eqn_CMathTable351GF_7.46} 
\sum_{n=0}^{\infty} n^m\ z^n & = \sum_{k=0}^{m} \gkpSII{m}{k} 
     \frac{k!\ z^k}{(1-z)^{k+1}} \\ 
\label{eqn_CM_7.46_gen_ident1} 
\sum_{n=0}^{\infty} (n-\alpha+1)^{m}\ z^n & = \sum_{k=0}^{m} \frac{
     [(1-z)^{k+1}] f_{m+1}}{(k+1)\ (1-z)^{k+1}} \\ 
\label{eqn_CM_7.46_gen_ident2} 
\sum_{n=0}^{\infty} (n-\alpha+1)^{m}\ z^n & = \sum_{j=1}^{m} 
     \FIIcfII{1 / \alpha}{m}{m+1-j} \frac{(-\alpha)^{j-1} (m-j)!}{ 
     (1-z)^{m+1-j}} \\ 
\label{eqn_CM_7.46_gen_ident3} 
\sum_{n=0}^{\infty} \phantom{an} (\alpha n+\beta)^{m}\ z^n & = 
     \sum_{n=0}^{\infty} \left(\sum_{k=0}^{m} \binom{m}{k} \alpha^k 
     \beta^{m-k} n^k\right) z^n 
\end{align} 
The classical \quotetext{\keyword{double}} generating functions 
enumerating the original Stirling number triangles are 
defined by the following forms as \cite[(7.54) and (7.55)]{GKP} 
\begin{equation*} 
\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \gkpSII{n}{m} w^m \frac{z^n}{n!} = 
     e^{w \left(e^{z}-1\right)} \quad \text{ and } \quad 
\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \gkpSI{n}{m} w^m \frac{z^n}{n!} = 
     \frac{1}{(1-z)^w} 
\end{equation*} 
and yield the generalizations to the $\alpha$-factorial triangle cases given 
in respective order by the next equations in 
\eqref{eqn_CM_7.54_GF_generalized} and \eqref{eqn_CM_7.55_GF_generalized}. 
\begin{align} 
\label{eqn_CM_7.54_GF_generalized} 
\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \FIIcfII{\alpha}{n}{m} 
     \frac{w^m z^n}{(n-1)!} & = wz e^{z} \left(e^{w( e^{\alpha z}-1)/
     \alpha}-1\right) \\ 
\label{eqn_CM_7.55_GF_generalized} 
\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \FcfII{\alpha}{n}{m} 
     \frac{w^m z^n}{n!} \phantom{-1} & = 
     \frac{(1-\alpha z)^{-(w+1)/\alpha}}{
     (\alpha-w-1)}\ \left((\alpha-1)(1-\alpha z)^{(w+1)/\alpha}+
     w (\alpha z-1)\right) 
\end{align} 
It follows from equation \eqref{eqn_CM_7.55_GF_generalized} that for 
$m, n \geq 1$ and for all $k \in \Nset$, the results of equations 
\eqref{eqn_partial_MF_exp_in_s_for_CM_7.55_gen} and 
\eqref{eqn_partial_MF_exp_in_s_for_CM_7.55_gen_V2} hold for the 
generalized triangle coefficients [cf.\ \sref{subSection_MF_expansion_polys}]. 
\begin{align} 
%% : "CM_7.55_generalized-working_V2.*"; 
\label{eqn_partial_MF_exp_in_s_for_CM_7.55_gen} 
\sum_{k=1}^{m} \FcfII{\alpha}{n}{k} \frac{s^{k-1}}{n!} & = [w^m z^n]\left( 
     \frac{(1-z \alpha )^{-\frac{1+sw}{\alpha }} \left((\alpha -1) 
     (1-z \alpha)^{\frac{1+sw}{\alpha }}-s w (1-\alpha z)\right)}{s (w-1) 
     (1-\alpha+sw)}\right) \\ 
\label{eqn_partial_MF_exp_in_s_for_CM_7.55_gen_V2} 
\sum_{k=1}^{n} \FcfII{\alpha}{n}{k} \frac{s^{k-1}}{n!} & = [z^n w^k]\left( 
     \frac{\left(1-\frac{\alpha z}{w}\right)^{-\frac{1+sw}{\alpha}} \left(w 
     \left(1-\frac{\alpha z}{w}\right)^{\frac{1+sw}{\alpha}}+\alpha z-w 
     \right)}{(w-1)(1-\alpha+sw)}\right) 
\end{align} 
The result of the identity in 
\eqref{eqn_partial_MF_exp_in_s_for_CM_7.55_gen_V2} 
corresponds to the coefficients on a prescribed diagonal index of the 
full generating function in equation 
\eqref{eqn_partial_MF_exp_in_s_for_CM_7.55_gen} 
\cite[cf.\ \Section 6.3]{ECV2}. Both of the identities are related to the 
$\alpha$-factorial function expansion polynomial properties discussed in 
\sref{subSection_MF_expansion_polys}. 

A generalization of the classical identity in \eqref{eqn_CMathTable351GF_7.49} 
\cite[(7.49)]{GKP} can be derived from the form of 
\eqref{eqn_Fa2_in_terms_of_S2_sum} and is given by equation 
\eqref{eqn_CM_7.49_GF_generalized}. 
\begin{align} 
\label{eqn_CMathTable351GF_7.49} 
 & \sum_{n=0}^{\infty} \FIIcfII{\phantom{\alpha}}{n}{m} \frac{z^n}{n!} = 
   \frac{(e^{z}-1)^m}{m!} \\ 
\label{eqn_CM_7.49_GF_generalized} 
 & \sum_{n=0}^{\infty} \FIIcfII{\alpha}{n}{m}\ \frac{z^n}{(n-1)!} = 
   \frac{\alpha^{1-m}}{(m-1)!}\ z e^{z} (e^{\alpha z}-1)^{m-1} 
\end{align} 
It can be shown from equation \eqref{Fa_Fa_BinomDeriv_ident} that for 
positive $m \in \mathbb{N}$ the following identity holds where the 
$\alpha$-factorial polynomial $\FaSigmaPoly{\alpha}{n}{x}$ is 
defined formally in \sref{subsubSection_FirstKindPolys_Intro_defs}. 
\begin{align*} 
\frac{(m-1)!}{(n-1)!}\ \FcfII{\alpha}{n+1}{m} & = \sum_{i=0}^{n-1} 
     \sum_{r=0}^{i} \left(\frac{(-1)^{n-1-i} \FaSigmaPoly{\alpha}{n-1-i}{\Nc} 
     \alpha^{r+1-m} (\Nc+1)^{r+1-m} (2-2\alpha)^{i-r}}{(r+1-m)!\ (i-r)!} 
     \right) \times \\ 
   & \phantom{=\sum\sum }\times \left(1+ 
     \frac{\alpha (\Nc+1) (m-1)}{(r+2-m)}\right) 
\end{align*} 
The identity provides the alternate generating function form for the analog of 
equation \eqref{eqn_CMathTable351GF_7.50} \cite[(7.50)]{GKP} given in 
equation \eqref{eqn_CM_7.50_GF_generalized_V1}. 
\begin{align} 
\label{eqn_CMathTable351GF_7.50} 
 & \sum_{n=0}^{\infty} \FcfII{\phantom{\alpha}}{n}{m} \frac{z^n}{n!} = 
   \frac{1}{m!} \Log\left(\frac{1}{1-z}\right)^m \\ 
\label{eqn_CM_7.50_GF_generalized_V1} 
 & \sum_{n=0}^{\infty} \FcfII{\alpha}{n+1}{m}\ \frac{z^n}{(n-1)!} = 
   \frac{\left(m-1+z\right)}{(m-1)!}\ z^{m-1} e^{z} 
   \left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)^{\Nc} 
\end{align} 
An alternate extension of \eqref{eqn_CMathTable351GF_7.50} \cite[(7.50)]{GKP} 
is given by equation \eqref{eqn_CM_7.50_GF_generalized} and 
is derived from the forms in \eqref{eqn_Szt_form_for_FaPolys2} and 
\eqref{eqn_Szt_form_for_FaPolys3} where $t := 1$ and 
$\mathcal{S}_{1}(\alpha z) = -\Log(1-\alpha z) / (\alpha z)$ 
[cf.\ \sref{pageref_subsubSection_Stirling_Polynomial_Forms}]. 
\begin{equation} 
\label{eqn_CM_7.50_GF_generalized} 
%% : "check_orig_3.29_for_CM7.50.*": 
\sum_{n=0}^{\infty} \FcfII{\alpha}{n}{m} \frac{z^n}{n!} 
     = \sum_{n=0}^{\infty} \left(\sum_{k=0}^n \frac{(-1)^{m+k}\ 
     \Log(1-\alpha z)^{m+k}\ (1-\alpha)^k}{\alpha^{m+k}\ (m+k)\ k!\ (m-1)!} 
     \right) z^n 
\end{equation} 

\subsection{Relations to Generalized Stirling Numbers} 
\label{Section_ConcludingRemarks} 

The triangles defined in 
\sref{subSection_Fa_Fa2_prelim_defs} may be compared to several of the 
treatments given in the referenced literature on Stirling numbers 
and the related properties of single factorial function expansions. 
Particular generalizations and variations on the 
classical Stirling number triangles are discussed in the references 
by Adelberg and others 
\cite{r-STIRLINGNUMS,D-STIRLINGNUMS,PROPSr-STIRSERIES,GENSNUMCVLFORMS} 
\cite[\Section 7]{ADELBERGA} \cite[\Section 3 and \Section 4]{CS} 
\cite[\Section 1.3 and \Section 2.4; cf.\ \Section 3.1]{K} 
\cite[\Section 4, (5.23), and (5.24)]{APPSEQSBETYPES} 
\cite[\Section 3]{CGSDEGENNUMSEQS}. 
Combinatorial interpretations and examples for the 
\keyword{generalized Stirling number} sets are discussed elsewhere by Lang 
\cite{JISLANGPDF}. Additionally, several Stirling number forms and 
properties are defined in 
terms of the differences of more generalized factorial functions in the 
work by Charalambides and Koutras \cite[\Section 4]{CK}. 

\subsubsection{Finite-Difference Properties of the 
               Non-Central Stirling Numbers} 
\label{subsubSection_NonCentralStirlingNumbersoftheFirstKind} 

The form of the \keyword{non-central Stirling numbers of the first kind} is 
discussed in Koutras' work \cite[\Section 1]{K}. 
This section explores analogous properties of the 
\eqref{Fa_rdef} triangle for the 
non-central Stirling numbers of the first kind. 
The properties given for the row sums of the 
non-central triangles \cite[Remark 3]{K} are analogous in form to 
\eqref{Fa_row_sum_ident} and \eqref{Fa_row_sum_ident_V2}. 
There are several additional properties for the 
non-central Stirling numbers of the first kind, 
$s_{a}(n;\ k)$, that are similar to the 
forms of \sref{subSection_FiniteDiffIdents} and result from expanding the 
non-central Stirling triangle recurrences in a manner 
similar to the derivation of the properties for the 
\eqref{Fa_rdef} triangle forms. 
 
To begin with, consider the recurrence form and corresponding conversion 
formula for the unsigned non-central Stirling numbers as follows 
\cite[cf.\ (1.2) and (1.3)]{K} [cf.\ \sref{subSection_Fa_Fa2_prelim_defs}]: 
\begin{align*} 
\bar{s}_a(n;\ k) & = (n+a-1)\ \bar{s}_a(n-1;\ k)+\bar{s}_a(n-1;\ k-1)+ 
     \Iverson{n = k = 0} \\ 
   & = (-1)^{n-k}\ s_{-a}(n;\ k) = \lvert\bar{s}_{-a}(n;\ k)\rvert. 
\end{align*} 
By expanding the recurrence definitions it is possible to express both the 
signed and unsigned triangles of the 
non-central Stirling numbers of the first kind through the 
next identities \cite[cf.\ \Section 1.2]{K}. 
\begin{align*} 
s_a(n;\ k) & = \sum_{j=0}^{n-k} \gkpSI{n}{n-j} \binom{n-j-1}{k-1} 
     (-1)^j a^{n-k-j} \\ 
s_a(n;\ k) & = \sum_{j=0}^{n-k} \binom{k-1+j}{k-1} (-1)^{n-k} n^j 
     s_{-a}(n;\ j+k) \\ 
\bar{s}_a(n;\ k) & = \sum_{j=0}^{n-k} \binom{k-1+j}{k-1} n^j 
     s_a(n;\ j+k) 
\end{align*} 
 
\label{subsubSection_NonCentralStirlingNumbersoftheSecondKind} 
The form of the \keyword{non-central Stirling numbers of the second kind} is 
also discussed in Koutras' work \cite[\Section 2]{K}. 
The following discussion briefly explores analogous properties of the 
\eqref{Fa2_rdef} triangle for the 
non-central Stirling numbers of the second kind. 
Similar to several of the identities in terms of powers of the input 
variables in \sref{subSection_FiniteDiffIdents} and to the 
properties noted for the non-central numbers of the first kind, the 
next pair of identities extend the properties given by Koutras 
\cite[cf.\ \Section 2.2]{K} for the 
non-central Stirling numbers of the second kind, $S_{a}(n;\ k)$. 
\begin{align*} 
S_a(n;\ k) & = \sum_{i=0}^{n-k} \gkpSII{n-i}{k}\binom{n-1}{i}\ (-a)^{i} \\ 
   & = \sum_{i=0}^{n-k} \sum_{j=0}^{n-k-i} \gkpSII{n-i-j}{k} 
     \binom{n-1}{i}\binom{n-i-1}{j}\ (-1)^{n-k-j}\ a^{i}\ (k+1)^{j} 
\end{align*} 

\subsubsection{Comparison to the Unified Generalizations of the 
               Classical Stirling Number Triangles} 
\label{subSection_GenStirlingNums_forms_and_diffs} 

The articles authored by Hsu et.\ al.\ \cite{HS,UNIFIEDST2} 
offer unified approaches to a number of separate 
generalizations of the classical Stirling cycle and subset triangles 
(first and second kinds, respectively). The work of Hsu and Shiue 
provides a more comprehensive discussion of unified properties that are 
analogous to the forms discussed in this article and so will be the 
focus of the Stirling number form comparisons addressed by this section. 
The $\alpha$-factorial triangles of the 
first kind in \eqref{Fa_rdef} and second kind in \eqref{Fa2_rdef} 
satisfy many similar properties to the 
\keyword{unified Stirling numbers}, though 
there are key distinctions in the forms from the treatment given in the 
references. 

To begin with, the particular manner that the first triangle \eqref{Fa_rdef} 
may be considered as a generalization of the classical set of 
Stirling numbers of the first kind is precisely the 
context of the author's first memorable encounter with these numbers: 
\emph{as factorial function expansions}. 
It appears that by considering the Stirling number generalizations 
as factorial function expansion coefficients some sense of the 
direct combinatorial meaning attached to the original triangle is obscured. 
In this case, the motivations of this article for generalizing the 
Stirling triangles gives an alternate, if separate, meaning to 
these triangles \cite[cf.\ \keyword{r-Stirling numbers}]{r-STIRLINGNUMS}. 

The motivation for constructing the triangles 
discussed in \sref{subSection_Fa_Fa2_prelim_defs} provides the 
non-dual triangle pair interpretations between the triangles of 
\eqref{Fa_rdef} and \eqref{Fa2_rdef} that yield the properties analogous 
to the classical forms offered by the last sections and also to 
the unified generalizations discussed by Hsu's work. The 
specific Stirling-number-like 
relation between the first \eqref{Fa_rdef} and 
second \eqref{Fa2_rdef} triangles defined by equations 
\eqref{FallingFact_gen_Fa_ident} and \eqref{xn_gen_Fa2_ident} is the 
key difference between the forms introduced by this article and the 
unified forms. Unlike both the classical Stirling triangles and the unified 
definitions, the \eqref{Fa_rdef} and \eqref{Fa2_rdef} triangles 
do \emph{not} conform to the typical \keyword{dual}, or 
\quotetext{\keyword{conjugate}}, relationship formed by 
the original triangles \cite[cf.\ (6.33)]{GKP} \cite{COMBIDSDUALS}. 
In contrast, the pair definition of $\{S^1,\ S^2\}$ given in the 
reference by Hsu \cite{HS} requires that the 
generalized Stirling triangles satisfy a symmetric relationship 
for the pair-based identities offered within that text. 
The distinction is particularly apparent when considering the 
relations of the separate $\alpha$-factorial polynomial sequences 
of the first and second kinds in 
\sref{subsubSection_JFactPoly_Initial_Properties} 
\cite[cf.\ Table 272; \Section 6.2 and \Section 6.5]{GKP}. 

In place of the unified set pairs, the following pair of identities may be 
defined in terms of the \keyword{Gould polynomials}, 
$G_n(x;\ a,\ 1) := x\Pochhammer{x-an-1}{n-1}$, also denoted 
$x^{[a;\ n]}$, through equation \eqref{eqn_GouldPoly_S1S2_pair_defs} 
\cite[cf.\ \Section 4.1]{CK} \cite[\Section 1.4]{UC} 
\cite[\Section 1]{PROPSr-STIRSERIES}. 
\begin{equation} 
\label{eqn_GouldPoly_S1S2_pair_defs} 
x^{[a;\ n]} = \sum_{k=0}^{n} t(n;\ k)\ x^k 
\quad \text{  and  } \quad 
x^{n} = \sum_{k=0}^{n} T(n;\ k)\ x^{[a;\ k]} 
\end{equation} 
As formulated in \sref{subSection_FiniteDiffIdents} and 
\sref{subSection_Fa_GF_form_idents}, the pair of triangles defined by 
this article in the forms of \eqref{Fa_rdef} and \eqref{Fa2_rdef}, 
as well as the alternate generalization suggested by 
equation \eqref{eqn_GouldPoly_S1S2_pair_defs}, 
still satisfy the orthogonality relations analogous to the unified form 
properties \cite[cf.\ \Section 5.1]{CK} \cite[\Section 1.3]{HS} and 
result in analogous enumerations compared to the 
generalized Stirling number sets that are defined in comparable forms by 
each of the unification articles \cite[Thm.\ 2; Remarks 1 and 2]{HS} 
\cite{UNIFIEDST2}. 
As noted in the text of \booktitle{The Umbral Calculus} 
\cite[\Section 1.4]{UC}, the case of the 
identities corresponding to the \keyword{central factorial polynomials}, 
denoted by the special case form of $x^{[n]} = x^{[-1/2;\ n]}$, is 
discussed in Riordan's book \cite{RIORDANUCREF}. 

For comparison, note that 
the recurrence relation \eqref{eqn_gkp_Y_rform_def} provides a more 
general form of the \eqref{Fa_rdef} and \eqref{Fa2_rdef} triangles, \emph{and} 
the unified Stirling number triangles \cite[Thm.\ 1]{HS}, as well 
additional combinatorial triangles of interest such as the first and 
\quotetext{second-order} Eulerian numbers noted in the 
discussion of \sref{subSection_Fa_Fa2_prelim_defs}. 
\begin{equation} 
\label{eqn_gkp_Y_rform_def} 
\gkpY{n}{k} = (\alpha n+\beta k+\gamma)\ \gkpY{n-1}{k}+ 
     (\alpha^{\prime} n+\beta^{\prime} k+\gamma^{\prime})\ \gkpY{n-1}{k-1} + 
     \Iverson{n = k = 0} 
\end{equation} 
A more thorough consideration of the general and special case forms of the 
triangles defined by \eqref{eqn_gkp_Y_rform_def} is handled in the 
excellent reference on the topic \cite[\Section 5 and \Section 6]{GKP}. 

\section{Symmetric Polynomial Transforms and Applications} 
\label{Section_Symm_poly_topics} 

Polynomial sequences and enumerative forms related to symmetric 
functions have a wide variety of combinatorial applications 
as discussed in several of the referenced works 
\cite[\Section 5]{r-STIRLINGNUMS} \cite{SYMMFNSPREC,SYMMFNSBINTYPE} 
\cite[cf.\ Prop. 2.1 (Proof 2) and Prop.\ 2.12]{ARXGENSTIRLING} \cite{ECV2}. 
For example, the classically defined 
Stirling number triangles may be defined in terms of, 
and have several properties related to, symmetric polynomials 
\cite[cf.\ \Section 2]{CS} \cite[cf.\ Prop.\ 2.1]{GENSNUMCVLFORMS}. 

The key and defining properties of the 
\eqref{Fa_rdef} triangle are related to the standard symmetric functions 
phrased by the definitions for the 
\keyword{elementary symmetric polynomial index transformations} in 
\sref{subSection_SymmFnTransforms_defs_and_properties}. 
The results offered in the next several sections are of particular interest 
since many of the forms progress from the 
finite-difference-based properties for the 
$\alpha$-factorial function expansions established by the 
previous section to fully analytic forms desired for the 
\emph{distinct} triangle expansion coefficients. 

\subsection{Elementary Symmetric Polynomial Index Transforms} 
\label{subSection_SymmFnTransforms_defs_and_properties} 

\subsubsection{Index Transform Preliminaries} 

Let the \keyword{elementary symmetric polynomial} function 
\cite[cf.\ \Section 2]{SYMMFNSPREC} be defined by 
\eqref{eqn_elem_symm_fn_e}. 
\begin{align} 
\label{eqn_elem_symm_fn_e} 
e_k(j) & := [z^k] \left(\prod_{m=0}^j (1+z\ x_m)\right) \\ 
\notag 
e_k(j) & \phantom{:}= \sum_{0 \leq < i_1 < \cdots < i_k \leq j} 
     x_{i_1} \cdots x_{i_k} + \Iverson{k = 0}\Iverson{j \geq 1}  
\end{align} 
The symmetric properties related to the definition of 
equation \eqref{eqn_elem_symm_fn_e} are considered as 
symmetric index transforms of the index inputs from the 
transformation function $x_m$ defined over the domain of natural numbers. 
Due to the distinction from the strictly symmetric polynomials 
given in terms of numbered distinct variables, 
the properties noted for the transformations do not necessarily 
result in \keyword{shift invariant} functions for the general case. 
The given symmetric interpretation can however be reconciled with the 
form of traditional symmetric polynomials by considering that the 
transformation function can be expanded in terms of the 
auxiliary notation for traditional elementary symmetric polynomials 
to obtain any desired shift invariant results as needed by 
applications external to the article \cite[cf.\ \Section 8]{SYMMFNSPREC}. 
 
The elementary symmetric function transforms considered by the 
applications in the context of this article are 
considered in a slightly more general form than the 
discussions of the next sections explicitly require. 
To begin with, define the special case index transform of the 
elementary symmetric function in \eqref{eqn_elem_symm_fn_e} by the 
form of equation \eqref{eqn_qkj_symm_tranform_fn_def}. 
\begin{equation} 
\label{eqn_qkj_symm_tranform_fn_def} 
q_k(j) := [z^k] \left(\prod_{m=0}^{j} (1+z\ (g+c m))\right) 
\end{equation} 
The symmetric index transform definition in 
\eqref{eqn_qkj_symm_tranform_fn_def} yields the 
exponential generating function in \eqref{eqn_qnm_EGF_form} that can be 
derived in terms of the constant parameters $g$ and $c$ that 
completely specify the transformation considered by the 
\eqref{eqn_qkj_symm_tranform_fn_def} form. 
\begin{equation} 
\label{eqn_qnm_EGF_form} 
\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \frac{q_n(m)}{m!}\ w^m z^n 
     = (1+g z)(1-c w z)^{\frac{-(1+(c+g) z)}{c z}} 
\end{equation} 
The definitions of equations \eqref{eqn_qkj_symm_tranform_fn_def} and 
\eqref{eqn_qnm_EGF_form} yield the following identities
where $\StirlingSigmaPoly{n}{x}$ is a 
Stirling (convolution) polynomial \cite[\Section 6.2]{GKP} and 
$B_n^{(a)}(z)$ is a generalized Bernoulli polynomial 
\cite{ADELBERGA,CARLITZC} \cite[\Section 2.2]{UC}. 
\begin{align*} 
q_n(m) & = \sum_{j=0}^{n} \binom{m+1-j}{n-j} 
     \gkpSI{m+1}{m+1-j}\ c^j\ g^{n-j} \\ 
q_n(m) & = \sum_{j=0}^{n} \left(c^j\ (m+1)\ \StirlingSigmaPoly{j}{m+1} 
     \left(1-\frac{j}{m+1}\right) \right) 
     \frac{(m+1)!\ g^{n-j}}{(m+1-n)!\ (n-j)!} \\ 
       & = \phantom{\sum_{j=0}^{n}} 
     \binom{m+1}{n}\ c^n\ B_n^{(m+2)}\left(m+1+\frac{g}{c}\right) 
\end{align*} 

\subsubsection{Generalized Triangle-Related Symmetric Index Transforms} 
\label{subsubSection_SymmPolys_Miz_defs} 

Consider the following identities phrased in similar recursive terms 
and form as the definition in equation \eqref{eqn_elem_symm_fn_e} where 
$\Pochhammer{a}{n} = \Gamma(a+n) / \Gamma(a)$ denotes the 
\keyword{Pochhammer symbol}, or \keyword{rising factorial} function. 
\begin{align} 
%% : "test_qj_eqn4.4.*": 
\label{qj_rform_idents1} 
q_j & := (1+(\gamma+\alpha-\alpha\ j) z)\ q_{j-1} + 
     \Iverson{j = 0} \\ 
\notag 
    & \phantom{:}= (-\alpha z)^{j-1} (1+\gamma z)\ 
      \Pochhammer{1-\frac{1+\gamma z}{\alpha z}}{j-1} \\ 
\notag 
\sum_{j=0}^{\infty} q_j\ w^j & \phantom{:}= \sum_{k=0}^{\infty}
      \sum_{j=0}^{\infty} \left(\sum_{i=0}^{k} \binom{i+j-k}{j-k} 
      \gkpSI{j}{i+j-k}\gamma^i\alpha^{k-i}\right) z^k w^j. 
\end{align} 
The next generating function over $z$ is given recursively by the form of 
equation \eqref{EQ.MGF} and is defined such that the 
requirement $[z^{i+1}] q_{n+i+1} = [z^n] M_i(z)$ holds in terms of the 
function of \eqref{qj_rform_idents1}. 
\begin{align} 
%% : "test_qj_eqn4.4.*": 
\label{EQ.MGF} 
M_i(z) & := \frac{(\gamma-\alpha i)M_{i-1}(z)-\alpha z 
         M^{\prime}_{i-1}(z)}{(1-z)}-\frac{(\alpha z-\gamma(1-z))}{(1-z)^3} 
         \Iverson{i = 0}+\frac{\Iverson{i = -1}}{(1-z)} 
\end{align} 
The definition of \eqref{EQ.MGF} is similar to the 
polynomial forms considered by M.\ Ward in the generating functions for the 
triangle of Stirling numbers of the first kind 
\cite[(2.41) and (2.5)]{MWARDFACTSUMS}. Ward's article considers 
polynomials over $z$ for each discrete index $i$ that are defined by the 
following equation. 
\begin{equation*} 
H_{i+1}(z) = (iz+i+1)\ H_{i}(z)+(1-z) z H^{\prime}_{i}(z) + \Iverson{i = 0} 
\end{equation*} 
While the context of the recurrence forms differs between the 
separate works, the forms of the $H_{i}(z)$ polynomial recurrence may 
eventually have uses in suggesting additional properties for the 
$M_i(z)$ form of equation \eqref{EQ.MGF}, if not in providing a 
full closed-form solution to the coefficient expressions related to the 
applications considered by this article. 

The imposed equivalence condition implies that the 
recurrence definition in \eqref{EQ.MGF} can be defined in terms of the 
Stirling numbers of the first kind through equation 
\eqref{EQ.MGF_S1_GF_eqn_form} 
\begin{equation} 
\label{EQ.MGF_S1_GF_eqn_form} 
M_i(z) = \sum_{n=0}^{\infty} \left(\sum_{k=0}^{i+1} \binom{n+k}{k} 
     (-1)^{n+i+1} \alpha^{i+1-k} \gamma^{k} \gkpSI{n+i+1}{n+k}\right) z^n, 
\end{equation} 
as well as in terms of the Stirling numbers 
on the series expansion index diagonals in the next equations of 
\eqref{eqn_Miz_S1_index_diagonal_form} and 
\eqref{eqn_Miz_SPoly_index_diagonal_form} \footnote{
     The \emph{Mathematica} package \texttt{\textbf{Stirling.m}} 
     provides additional recursive forms for similar sums involving the 
     Stirling or Eulerian numbers and hypergeometric terms through the 
     routine \texttt{\textbf{FindRec}} implemented by the 
     software package cited in the references section \cite{STIRLINGMMPKG}. 
}. 
%% : "Miz_Fa_triangle_idents-2009.07.22-V2.*": 
\begin{align} 
\label{eqn_Miz_S1_index_diagonal_form} 
[z^k] M_{i-k}(z) & = \sum_{j=0}^{i-k+1} \left(\gkpSI{i+1}{j+k} \binom{j+k}{k} 
     (-\alpha)^{i-k+1-j}\ \gamma^{j}\right)-\Iverson{k = i + 1} \\ 
\label{eqn_Miz_SPoly_index_diagonal_form} 
[z^k] M_{i-k}(z) & = \sum_{j=0}^{i-k+1} \left(\frac{(j+k)\ 
     (-\alpha)^{i-k+1-j}\ \gamma^j\ i!}{k!\ j!}\ 
     \StirlingSigmaPoly{i-k+1-j}{i+1}\right) -\Iverson{k = i + 1} 
\end{align} 
Let the shorthand 
$[z^n] M_i(z) := m_i(n)$ denote the coefficients that define the 
series expansion of the generating function \eqref{EQ.MGF}. 
Then for the transform function $Q(j) := (\gamma-\alpha j)$ 
corresponding to $g := \gamma$ and $c := -\alpha$ in equation 
\eqref{eqn_qkj_symm_tranform_fn_def}, the 
following symmetric index transform identities also hold for the coefficient 
forms over the indices $n \in \mathbb{N}$ and 
$i \in [-1,\ \infty] \subseteq \mathbb{Z}$. 
\begin{align*} 
%% : "Miz_verify_idents.*": 
%[z^n] M_i(z) & = \sum_{k=0}^{n} Q(k+i)\ m_{i-1}(k)+ 
%     \sum_{k=0}^{n} Q(k) \Iverson{i = 0} + \Iverson{i = -1} = q_{i+1}(n+i) 
[z^n] M_i(z) & = \sum_{k=0}^{n} Q(k+i)\ m_{i-1}(k)+\Iverson{i = -1} 
                = q_{i+1}(n+i) 
\end{align*} 
The specific symmetric-index-based property for the 
\eqref{Fa_rdef} triangle is given by equation \eqref{eqn_Miz_q_ident} where 
$\gamma := (\alpha \Nc+1-2\alpha)$ \cite[cf.\ \Section 2]{MWARDFACTSUMS}. 
\begin{equation} 
\label{eqn_Miz_q_ident} 
\FcfII{\alpha}{n}{k} = [z^{k-2}]M_{n-k-1}(z)+[z^{k-1}]M_{n-k-2}(z) 
\end{equation} 
Additionally, the identities of \eqref{eqn_Miz_Fa_ident_V1}, 
\eqref{eqn_Miz_Fa_ident_V2}, and \eqref{eqn_Miz_q_ident_GF_coeff} hold for the 
generalized triangle coefficients as given in terms of equation 
\eqref{eqn_Miz_q_ident} when $n \in [3,\ \infty) \subseteq \Nset$ and 
where $\StirlingSigmaPoly{n}{x}$ denotes the 
sequence of Stirling polynomials \cite[\Section 6.2]{GKP}. 
%% : "Miz_Fa_triangle_idents-2009.07.22.*": 
\begin{align} 
\label{eqn_Miz_Fa_ident_V1} 
\FcfII{\alpha}{n}{k} \quad & = \sum_{0 \leq i \leq 1} \left(\sum_{j=0}^{n-k-i} 
     \gkpSI{n-2}{j+k+i-2} \binom{j+k+i-2}{k+i-2} \frac{(-\alpha)^{n-k-i-j}\ 
     \gamma^{j}}{n!}\right) \\ 
\label{eqn_Miz_Fa_ident_V2} 
   & = \sum_{0 \leq i \leq 1} \left(\sum_{j=0}^{n-k-i} \frac{(j+k+i-2)\ 
     (-\alpha)^{n-k-i-j}\ \gamma^{j}}{n (n-1)\ (k+i-2)!\ j!}\ 
     \StirlingSigmaPoly{n-k-i-j}{n-2}\right) \\ 
\label{eqn_Miz_q_ident_GF_coeff} 
   & = [z^n] \left(\frac{z^k \left((k-1) (\alpha (n-2)+2) z 
     +(\alpha (n-2)+1) z^2+(k-1)(k-2)\right)}{n(n-1)(n-2)\ (\alpha z)^{2-n} 
     \left(e^{\alpha z}-1\right)^{n-2}e^{-(\alpha (n-2)+1) z} \ 
     \Gamma(k)}\right) 
\end{align} 
The last listing of the identities given for the 
\eqref{Fa_rdef} triangle coefficients are actually 
special case forms of the $M_i(z)$ series coefficients on the index 
diagonals defined by the forms of equations 
\eqref{eqn_Miz_S1_index_diagonal_form} and 
\eqref{eqn_Miz_SPoly_index_diagonal_form}. 

\subsection{Properties and Enumeration Results for the 
            Generalized $j$-Factorial Function Expansion Coefficient Triangles} 

\subsubsection{Initial Properties} 
\label{subSection_init_Fa_symm_idents} 

Let the function $Q_{\alpha}(j) := (1-\alpha(j+2))$ and define the 
next symmetric index transform by the following equation 
corresponding to the case where $x_m := Q_{\alpha}(m)$ in the 
product expansion for the function $e_k(j)$ of equation 
\eqref{eqn_elem_symm_fn_e}. 
\begin{equation*} 
q_k^{\alpha}(j) := [z^k] \left(\prod_{m=0}^j 
     (1+z\ Q_{\alpha}(m))\right) 
\end{equation*} 
In general, for integer index $n \geq 3$, the 
triangle in \eqref{Fa_rdef} can be defined in terms of the given expansion 
forms and symmetric properties through the following equations. 
\begin{align} 
\notag 
p_n^{\alpha}(0) & := \prod_{m=0}^{n-3} (\alpha n+1-(m+2) \alpha) 
    =  \sum_{k=0}^{n-2} \FcfII{\alpha}{n-1}{k+1}\ 
     (-1)^{n}\ (2\alpha-\alpha n-2)^{k} \\ 
\label{eqn_pn0_general_exp_form} 
p_n^{\alpha}(0) & \phantom{:}= \prod_{m=0}^{n-3} (\alpha n+Q_{\alpha}(m)) = 
     \sum_{k=0}^{n-2} q^{\alpha}_{n-2-k}(n-3)\ 
     (\alpha n)^k \\ 
\label{eqn_pnj_general_exp_form} 
p_n^{\alpha}(j) & \phantom{:}= \sum_{i=j}^{n-2} \binom{i}{j} 
     \frac{[n^i]\ p_n^{\alpha}(0)}{\alpha^j}\ n^{i-j} % \\ 
   = \sum_{i=j}^{n-2} \binom{i}{j}\ q^{\alpha}_{n-2-i}(n-3)\ 
     (\alpha n)^{i-j} 
\end{align} 
The coefficients of $p_n^{\alpha}(0)$ in \eqref{eqn_pnj_general_exp_form} 
are given by the next form of equation \eqref{eqn_pn0_coeff_of_ni}. 
\begin{align} 
\label{eqn_pn0_coeff_of_ni} 
[n^i]\ p_n^{\alpha}(0) & = \sum_{k=i}^{n-2} \FcfII{\alpha}{n-1}{k+1} 
     \binom{k}{i}\ (-1)^{n+i}\ \alpha^{i}\ (2\alpha-2)^{k-i} 
\end{align} 
An identity for $p_n^{\alpha}(j)$ resulting from the 
coefficient form of the last equation is given by 
\eqref{eqn_pnj_form_pn0_coeff_of_ni}. 
\begin{align} 
\label{eqn_pnj_form_pn0_coeff_of_ni} 
p_n^{\alpha}(j) & = \sum_{i=0}^{n-2-j} \sum_{k=0}^{n-2} 
     \FcfII{\alpha}{n-1}{k+1}\ \binom{i+j}{j} \binom{k}{i+j}\ (-1)^{n-2-j-i}\ 
     \alpha^{i}\ n^{i}\ (2\alpha-2)^{k-j-i} 
\end{align} 
The symmetric-based form for the triangle coefficients of 
\eqref{Fa_rdef} then results in the next expression given in equation 
\eqref{eqn_Fa_pnj_sum_form}. 
\begin{equation} 
\label{eqn_Fa_pnj_sum_form} 
\FcfII{\alpha}{n}{k} = p_n^{\alpha}(k-1)+p_n^{\alpha}(k-2)
\end{equation} 

The same construct involved in the formulation of the so termed 
\quotetext{binomial derivative} identities in 
\sref{subSection_FiniteDiffIdents} applies to the identities given in this 
section. Observe that if the forms are treated as polynomials in $n$ 
with discrete powers independent of the input variable, equation 
\eqref{eqn_pnj_general_exp_form} may be considered as a 
multiple derivative with respect to $n$. However, note that 
differentiating the same polynomial with the degree of the 
lead term defined in terms if the input variable as the expression $n^{n-k}$ 
gives a result that differs from the desired forms considered by the 
finite-difference approaches in this context. This distinction 
may be summarized by comparing the form of 
$\frac{\partial}{\partial x}\left[x^{p}\right]$ versus the 
form that results from the variable-dependent expression in 
evaluating $\frac{\partial}{\partial x}\left[x^{x-k}\right]$ 
for fixed $p, k \in \mathbb{N}$. 

\subsubsection{An Exponential Generating Function} 
\label{subSection_symm_EGF} 

Let the function $R_{\alpha}(j) := (\alpha \Nc+1-(j+2) \alpha)$ and 
define the following symmetric index transform corresponding to the 
case where $x_m := R_{\alpha}(m)$ in the function $e_k(j)$ of equation 
\eqref{eqn_elem_symm_fn_e}. 
\begin{equation*} 
r_k^{\alpha}(j) := [z^k] \left(\prod_{m=0}^j 
     (1+z\ R_{\alpha}(m))\right) 
\end{equation*} 
The form of of the triangle coefficients in \eqref{Fa_rdef} is related to the 
given index transform through the next equation. 
\begin{equation*} 
\FcfII{\alpha}{n}{k} = r^{\alpha}_{n-1-k}(n-3)+r^{\alpha}_{n-k}(n-3) 
\end{equation*} 
Next, let the function $\QPS: \Nset \rightarrow \Zset$ be defined and 
enumerated as in the form of equations \eqref{eqn_QPS_def_V1} and 
\eqref{eqn_QPS_def_V2}. 
\begin{align} 
\label{eqn_QPS_def_V1} 
\QPS(m) & := z^{j+2}\ (z+1)\ \prod_{j=0}^m (1+R_{\alpha}(j) z^{-1}) \\ 
\label{eqn_QPS_def_V2} 
\QPS(m) & \phantom{:}= \frac{(-1)^{m+1} \alpha^{m+1} z(z+1)
     \Gamma(m+3-\Nc-\frac{z+1}{\alpha})}{
     \Gamma(2-\Nc-\frac{z+1}{\alpha})} 
\end{align} 
Additionally, define the exponential generating function for the function 
$\QPS(m)$ by equation \eqref{eqn_QPS_EGF_enum}. 
\begin{equation} 
\label{eqn_QPS_EGF_enum} 
\widehat{\QPS}(w;\ z) := \sum_{j=0}^{\infty} \frac{\QPS(j)}{j!}\ w^j 
     = z (z+1) (z+1+(\Nc-2)\ \alpha)\ 
     (1+\alpha w)^{\Nc-3+\frac{z+1}{\alpha}} 
\end{equation} 
The form of the \eqref{Fa_rdef} triangle coefficients resulting from the 
terms in the series expansion of the exponential generating function 
can then be expressed by equation 
\eqref{Fa_rdef_in_terms_of_QPS_and_QPS_EGF_coeffs}.  
\begin{equation} 
\label{Fa_rdef_in_terms_of_QPS_and_QPS_EGF_coeffs} 
\FcfII{\alpha}{n}{k} = [z^k] \QPS(n-3) 
     = (n-3)!\ [z^k w^{n-3}] \widehat{\QPS}(w;\ z) 
\end{equation} 
The coefficients in the series expansion of $\widehat{\QPS}(w;\ z)$ 
involved in evaluating the form of the equation in 
\eqref{Fa_rdef_in_terms_of_QPS_and_QPS_EGF_coeffs} are given by 
\begin{equation} 
\label{eqn_QPS_EGF_coeff_wmzn} 
[w^m z^n] \widehat{\QPS}(w;\ z) = 
     \sum_{i+j+k= n} q_1(i;\ n) q_2(j;\ n) q_3(k;\ n) 
\end{equation} 
where the functions $q_i(m;\ n)$ in equation \eqref{eqn_QPS_EGF_coeff_wmzn} 
are defined as follows \cite[cf.\ \Section 8]{ARXONASYM}: 
\begin{align*} 
q_1(m;\ n) = & [w^m] \Log(1+\alpha w)^{n-3} \\ 
q_2(m;\ n) = & \frac{\alpha^{m+1-n}}{\Gamma(n)}\ 
             \binom{\Nc-3+\frac{1}{\alpha}}{m} \\ 
q_3(m;\ n) = & \frac{2\alpha^m (\alpha \Nc-2\alpha+1) 
     \Gamma(m-\Mc-1) (-\PolyGamma{0}{m-\Mc-1}+\PolyGamma{0}{1-\Mc)}+1)}{
     \Gamma(m+1)\ \Gamma(1-\Mc)} \\ 
   & +\frac{(-1)^{m+1}\alpha^{m+2}}{m}\ (n-1)(\Nc-2). 
\end{align*} 
Note that the powers of the natural logarithm function in $q_1(m;\ n)$ may be 
considered in terms of standard Stirling number 
properties through equation \eqref{eqn_CMathTable351GF_7.50} 
for negative integer exponents, the duality identity given by 
$\gkpSI{n}{k} = \gkpSII{-k}{-n}$ \cite[(6.33)]{GKP}, and 
the polynomial identities of Table $272$ as listed in the 
\booktitle{Concrete Mathematics} reference \cite[\Section 6.2]{GKP}. 

\label{footnote_Nc_Mc_notation} 
The discussion on fixed variable notation outlined in 
\sref{subSection_Intro_notations} is also relevant to the forms 
defined by the section. In particular, one notable 
caveat point of the fixed variable notation appears in evaluating inputs to the 
gamma function, $\Gamma(z)$, and 
\keyword{polygamma function}, $\PolyGamma{k}{z}$, in the 
function $q_3(m;\ n)$ defined through equation \eqref{eqn_QPS_EGF_coeff_wmzn}. 
The fixed parameter $\Mc$ in this case \emph{cannot} be 
evaluated at exact integer values 
until after corresponding non-constant variable index $m$ 
has completely defined the 
symbolic expansions in the identity. A computer-based 
application such as \emph{Mathematica} can be used to verify the special 
utility of the expansions that result from this example case of the 
fixed variable notation for the index $m$ in terms of the parameter $\Mc$. 

\subsubsection{Other Forms of the Generalized Triangle Coefficients} 
\label{subsubSection_SymmTransforms_pn_product_function_forms} 

Consider the product function definition and 
expansion identity given by the form of the following equations. 
\begin{align*} 
\widetilde{p}(n) & := \frac{(-1)^n \alpha^{n-2} \gamma}{n!}\ 
     \prod_{m=1}^{n-2} \left(m-\frac{\gamma}{\alpha}\right) = 
     \frac{(-1)^{n+1} \alpha^{n-1} \Gamma(n-1-\frac{\gamma}{\alpha})}{ 
     \Gamma\left(-\frac{\gamma}{\alpha}\right) \Gamma(n+1)} \\ 
\widetilde{p}(n) & \phantom{:}= \sum_{j=1}^n \gkpSI{n-1}{j} 
     \frac{(-1)^{n-1-j}}{n!} \alpha^{n-1-j} \gamma^j 
\end{align*} 
The $k^{th}$ derivatives of the function $\widetilde{p}(n)$ can be 
expressed by the following forms where $\StirlingSigmaPoly{n}{x}$ denotes the 
sequence of Stirling polynomials [cf.\ \sref{Section_MF_Fa_polynomials}] and 
$E_n(x) = \int_{1}^{\infty} {e^{-x t} t^{-n}}dt$ denotes the 
\keyword{exponential integral} function. 
\begin{align*} 
\frac{\partial^{k}}{\partial \gamma^{k}} \left[
     \frac{\widetilde{p}(n)}{k!}\right] & \phantom{:}= 
     \frac{\alpha^{n-1}}{n!}\ \sum_{j=1}^n \binom{j}{k}\gkpSI{n-1}{j} 
     \frac{(-1)^{n-1-j}\ \gamma^{j-k}}{\alpha^j} \\ 
   & \phantom{:}= \sum_{j=0}^{n-2} \frac{(j+1)\ \gamma^{j+1-k}}{k!\ 
     (j+1-k)!}\ \frac{(-\alpha)^{n-2-j}}{n}\ \StirlingSigmaPoly{n-2-j}{n-1} \\ 
   & \phantom{:}= [z^n] \left(\frac{
     \Log (1+\alpha z) E_{-k}\left((1+\frac{\gamma}{\alpha}) \Log
      \left(\frac{1}{1+\alpha z}\right)\right)+\alpha z (1+\alpha z)^{ 
      \frac{\gamma}{\alpha}}}{(n-1) \Log(1+\alpha z)^{-k}\ \alpha^{k+1} 
      \Gamma (k+1)}\right)+\KDelta{n}{1}\KDelta{k}{0} 
\end{align*} 
Next, consider the generating function for $\widetilde{p}(n)$ 
defined by the form of the following equation. 
\begin{equation*} 
P(z) := \sum_{n=0}^{\infty} \widetilde{p}(n) z^n = 
     \frac{(1+\alpha z)^{1+\frac{\gamma}{\alpha}}}{(\alpha+\gamma)} 
\end{equation*} 
The form of the $k^{th}$ derivatives of $\widetilde{p}(n)$ can then be 
enumerated precisely by considering the properties of the derivatives of the 
ordinary generating function $P(z)$ as given in terms of the equations below 
\cite[cf.\ \Section 4.4]{UC}: 
\begin{align*} 
\frac{\partial^{(k)}}{\partial \gamma^{(k)}}\left[\frac{\widetilde{p}(n)}{k!} 
     \right] & := \left[z^n\right]\ \frac{\partial^{(k)}}{\partial 
     \gamma^{(k)}}\left[\frac{P(z)}{\Gamma(k+1)}\right] \\ 
   & \phantom{:}= \left[z^n\right] \left(\sum_{j=0}^{k} \frac{(-1)^{k+j} 
     \Log(1+\alpha z)^{j}}{j!\ \alpha^{j} (\alpha+\gamma)^{k+1-j}}\ 
     (1+\alpha z)^{1+\frac{\gamma}{\alpha}}\right) \\ 
   & \phantom{:}= \left[z^n\right] \left(\frac{(-1)^{k}\ 
     \Gamma(k+1;\ -(1+\frac{\gamma}{\alpha}) 
     \Log(1+\alpha z))}{(\alpha+\gamma)^{k+1} \Gamma(k+1)}\right). 
\end{align*} 
For positive $k \in \mathbb{N}$, the 
$k^{th}$ derivative of $P(z)$, denoted $d_k$, 
satisfies the recurrence relation given in equation 
\eqref{eqn_dk_Pz_kth_deriv} and 
can be enumerated by the corresponding ordinary generating function in 
equation \eqref{eqn_Dt_GF_form}. 
\begin{align} 
\label{eqn_dk_Pz_kth_deriv} 
 & \phantom{\sum_{k=1}^{\infty}} d_k \phantom{\ t^k} = \frac{-k}{(\alpha+\gamma)} d_{k-1}+
     \frac{(1+\alpha z)^{1+\frac{\gamma}{\alpha}} \Log(1+\alpha z)^{k}}{ 
     \alpha^{k} (\alpha+\gamma)}-\frac{(1+\alpha z)^{1+\frac{\gamma}{\alpha}} 
     }{(\alpha+\gamma)^2} \Iverson{k = 1} \\ 
\notag 
 & \sum_{k=1}^{\infty} d_k\ t^k = 
\frac{e^{-(t-1) (\alpha+\gamma)/t}}{\alpha t (\alpha+\gamma)^2} 
     \Bigl((1+\alpha z)^{\frac{\alpha+\gamma}{\alpha}} ((\alpha+\gamma) 
     \Log(1+\alpha z)-\alpha)+\alpha e^{\alpha+\gamma} (\alpha+\gamma)^2\ 
     \times \\ 
\label{eqn_Dt_GF_form} 
 & \quad \qquad \times \int_1^t \frac{e^{-\frac{\alpha+\gamma}{t}} 
     (1+\alpha z)^{\frac{\alpha+\gamma}{\alpha}} \Log(1+\alpha z) 
     (\Log(1+\alpha z) t-\Log(1+\alpha z) t+\alpha)}{\alpha (\alpha- 
     \Log(1+\alpha z) t)} \, dt\Bigr) 
\end{align} 
Identities for special case columns of the generalized triangle 
coefficients in \eqref{Fa_rdef} 
may then be derived by computing the iterated derivatives of $\widetilde{p}(n)$ 
from equation \eqref{eqn_S1_pfn_spec_case} and in the coefficient form of 
equation \eqref{eqn_Fa_pGF_Deriv_closedform_ident} where the second form is 
derived from the recursive definition provided by 
\eqref{eqn_dk_Pz_kth_deriv} 
\cite[cf.\ Prop.\ 2; Prop.\ 3; and (17)]{STIRESUMS}. 
\begin{align} 
\label{eqn_S1_pfn_spec_case} 
\frac{1}{n!}\ \FcfII{\alpha}{n+1}{k} & = \frac{\partial^{k-1}}{
     \partial \gamma^{k-1}} \left[\frac{\widetilde{p}(n)}{(k-1)!}\right] + 
     \frac{\partial^{k-2}}{\partial \gamma^{k-2}} 
     \left[\frac{\widetilde{p}(n)}{(k-2)!}\right] \\ 
\label{eqn_Fa_pGF_Deriv_closedform_ident} 
%% : "Fa_Deriv_pGF_triangle_coeff_ExpInt_closedform.*": 
  & = [z^n]\left(\frac{n \Log(1+\alpha z) E_{1-k} 
    \left(-(n+\frac{1}{\alpha}) \Log(1+\alpha z)\right)+ 
    (1+\alpha z)^{\frac{1}{\alpha}+n}}{\alpha^{k-1} \Log(1+\alpha z)^{1-k}\ 
    \Gamma(k)}\right) 
\end{align} 
The parameter $\gamma$ in \eqref{eqn_S1_pfn_spec_case} must again be 
evaluated after the symbolic computation of the partial derivative terms and in 
terms of the substitution defined by $\gamma := (\alpha \Nc+1-\alpha)$. 

The definition of equation \eqref{eqn_S1_pfn_spec_case} may also be used to 
construct additional \quotetext{\keyword{horizontal}} and 
\quotetext{\keyword{vertical}} recurrence relations for the 
\eqref{Fa_rdef} triangle coefficients that are analogous to the forms 
defined for the classical Stirling number triangles by the 
work of Charalambides and Singh \cite[p.\ 2543]{CS}. The triangle identity in 
equation \eqref{eqn_S1_pfn_spec_case} suggests a systematic procedure for 
recursively computing more specific closed-form expressions for fixed 
columns of the \eqref{Fa_rdef} triangle in terms of the gamma, 
incomplete gamma, polygamma, exponential integral, and related functions. 
In particular, these closed-forms include the special cases for the 
classically-defined triangle of Stirling numbers of the first kind as 
specifically detailed in several applications related to sequences of 
\keyword{harmonic numbers} that are discussed next and in 
\sref{subsubSection_HarmonicNumber_Idents}. 

\subsection{Harmonic Number Applications} 
\label{subSection_SymmTransform_and_HarmonicNumber_Apps} 

\subsubsection{Generating Functions and Other Identities for the 
               Sequence of First-Order Harmonic Numbers} 

The sequence of \keyword{first-order harmonic numbers} 
\citeseqnumII{A001008}{A002805} is defined classically through the 
identity \cite{GKP} 
\begin{equation*} 
H_n := H_{n-1}+\frac{1}{n}+\Iverson{n = 1} = 
     \sum_{k=1}^n \frac{1}{k}. 
\end{equation*} 
The first-order harmonic numbers also satisfy the 
following properties in terms of the 
symmetric-index-transform-related function forms in 
\sref{subsubSection_SymmPolys_Miz_defs} where 
$\StirlingSigmaPoly{n}{x}$ denotes the sequence of Stirling polynomials 
\cite{CVLPOLYS} \cite[\Section 6.2]{GKP} 
[cf.\ \sref{Section_MF_Fa_polynomials}]. 
\begin{align*} 
H_n & = \frac{1}{n!} \gkpSI{n+1}{2} = \frac{1}{n!} 
      \left([z^0]M_{n-2}(z)+[z^1]M_{n-3}(z)\right) \\ 
    & = \frac{1}{n!} \sum_{k=1}^{n-1} \gkpSI{n-1}{k} (-1)^{n-1-k} 
      \alpha^{n-1-k} \gamma^{k-1}\ (\gamma+k) \\ 
    & = \phantom{\frac{1}{n!}} \sum_{k=0}^{n-2} 
      \frac{(-\alpha)^{n-2-k}}{n} \StirlingSigmaPoly{n-2-k}{n-1} 
      (1+\gamma+k) \frac{\gamma^{k}}{k!} 
\end{align*} 
The harmonic number sequence corresponds to the case of the 
\eqref{Fa_rdef} triangle properties for 
Stirling numbers of the first kind when $\alpha := 1$ and 
where $\gamma := \alpha(\Nc+1)+1-2\alpha = \Nc$. 

The form of the last property for $H_n$ provides an enumeration 
for the harmonic number sequence that is derived from the 
ordinary generating function for the 
Stirling (convolution) polynomial sequence from 
\eqref{eqn_StirlingSigmaPoly_enum_CM_6.50} \cite[(6.50)]{GKP} and 
that results in a series expansion for that enumeration 
expressed in terms of the \keyword{N\"{o}rlund polynomials} $B_n^{(x)}$ as 
given by the following equations 
[cf.\ \sref{subsubSection_GenBernoulliPoly_ident_forms};\ 
\eqref{eqn_UC_gen_BP_Hn_ident} and \eqref{eqn_CM_7.50_GF_generalized_V1}]. 
\begin{align*} 
H_{S}(z) & := \sum_{n=1}^{\infty} H_n z^n = 
     z+\frac{(-1)^{\Nc}\ z^{\Nc+1} \left(e^z-1\right)\ (\Nc\ z+\Nc+1) 
     \left(e^{-z}-1\right)^{-\Nc}}{(\Nc-1)\ \Nc} \\ 
[z^n] H_{S}(z) & := H_n = \Iverson{n = 1} 
     +\frac{(-1)^{n-2}\ (n+1) B_{n-2}^{(n)}}{n!}+\sum_{j=3}^{n} 
     \frac{(-1)^{n-j}\ (1+nj)}{n(n-1) (j-1)! (n-j)!} B_{n-j}^{(n)} 
\end{align*} 

It follows from the symmetric identities that the 
harmonic number sequence is enumerated by the 
ordinary generating function in \eqref{eqn_Hn_OGF}, 
the exponential generating functions in equations \eqref{eqn_Hn_EGF_V1} and 
\eqref{eqn_Hn_EGF_V2}, and by the \keyword{doubly}, or \keyword{twice}, 
exponential generating function in \eqref{eqn_Hn_double_EGF}, 
where $L_n(x)$ denotes the sequence of 
\keyword{Laguerre polynomials} \cite[\Section 3.1]{UC}, 
$\PolyGamma{0}{z}$ denotes the polygamma function 
(\keyword{digamma function}), $\gamma_{\E}$ denotes the 
\keyword{Euler-Mascheroni constant}, and $I_k(z)$ denotes the 
\keyword{modified Bessel function of the first kind} 
\cite[\Section 5.5]{GKP} \cite[cf.\ \Section 1.7]{UC}. 
%% : "working_S1_Hn_sp_case-Hz_EGF_series_coeff_forms.*": 
\begin{align} 
\label{eqn_Hn_OGF} 
H(z) := \sum_{n=1}^{\infty} H_n z^n & = 
     \frac{(1+z)^{\Nc+1} ((\Nc+1) \Log(1+z)+\Nc)}{(\Nc+1)^2} \\ 
\label{eqn_Hn_EGF_V1} 
\widehat{H}(z) := \sum_{n=1}^{\infty} \frac{H_n}{n!} z^n & = 
     \frac{-\left(4\Nc+4 (\Nc+1)^2 z+(\Nc+1)^2 z^2\right)}{4 (\Nc+1)^2} \\ 
\notag & \phantom{=} 
     +\frac{\Nc}{(\Nc+1)^2} L_{\Nc+1}(-z)+\frac{1}{(\Nc+1)} 
     \frac{\partial}{\partial \Nc}L_{\Nc+1}(-z) \\ 
\label{eqn_Hn_EGF_V2} 
   & = \frac{e^z (z+1) (\gamma_{\E} \Nc+\Nc \PolyGamma{0}{\Nc+2} 
     +\PolyGamma{0}{\Nc+2}+\gamma_{\E}-1)}{(\Nc+1)^2} \\ 
\label{eqn_Hn_double_EGF} 
\widetilde{H}(z) := \sum_{n=1}^{\infty} \frac{H_n}{(n!)^2} z^n & = 
     \frac{(\PolyGamma{0}{\Nc+2}+\gamma_{\E}-1+(\PolyGamma{0}{\Nc+2}+ 
     \gamma_{\E}) \Nc)}{(\Nc+1)^2 \left(I_0\left(2 \sqrt{z}\right)+ 
     \sqrt{z}I_1\left(2 \sqrt{z}\right)\right)^{-1}}
\end{align} 
The forms of equations \eqref{eqn_Hn_EGF_V2} and \eqref{eqn_Hn_double_EGF} are 
derived from \eqref{eqn_Hn_EGF_V1} by noting the particular 
series expansion forms for the Laguerre polynomials as follows: 
\begin{align*} 
[z^n] L_{\Nc+1}(-z) & = \frac{(-1)^n\ \Nc \Gamma(n-\Nc-1)}{(\Nc+1)^2\ 
     \Gamma(n+1)\ \Gamma(-\Nc-1)} \\ 
   & = \frac{(\Nc+1)}{(n!)^2} \left(\prod_{j=0}^{n-2} (\Nc-j) 
     \right) \Iverson{n \geq 1} +\Iverson{n = 0} \\ 
   & = \frac{\Gamma(\Nc+2)}{(n!)^2} \Iverson{n \geq 1} + 
     \Iverson{n = 0} \\ 
[z^n] \frac{\partial}{\partial \Nc}\left[L_{\Nc+1}(-z)\right] & = 
     \frac{(-1)^{n+1}\ \Gamma(n-\Nc-1) (\PolyGamma{0}{-\Nc-1} 
     \PolyGamma{0}{n-\Nc-1})}{\Gamma(n+1)\ \Gamma(-\Nc)} \\ 
   & = \frac{1-(n+1) \PolyGamma{0}{n+1}}{\Gamma(n+1)}. 
\end{align*} 

Given the form of equation \eqref{eqn_Hn_OGF}, it is possible to 
formulate additional identities for the first-order harmonic numbers from the 
series coefficients in that enumeration of the sequence. 
Specifically, for $k \in [1,\ \infty) \subseteq \Nset$ and 
for the index $\Nc \mapsto \Kc$, the following properties are 
derived from the ordinary generating function for $H(z)$. 
%% : "working_S1_Hn_sp_case-Hz_OGF_series_coeff_forms.*": 
\begin{align*} 
H_k := [z^k] H(z) & = \sum_{j=1}^{k} \binom{\Kc+1}{k-j} 
     \frac{(-1)^{j+1}}{j\ (\Kc+1)} + \frac{\Kc}{(\Kc+1)^2} \binom{\Kc+1}{k} \\ 
   & = \sum_{j=1}^{k} \binom{k+1}{k-j} \frac{(-1)^{j+1}}{j\ (k+1)} 
     + \frac{k}{k+1} \\ 
   & = \gamma_{E}-\frac{1}{k+1}+\PolyGamma{0}{k+2} 
\end{align*} 
The article by Adamchik also notes the related identity for the 
first-order harmonic numbers in terms of the 
Stirling triangle coefficients given by the next equation 
\cite[\Section 4]{STIRESUMS} \cite[cf.\ (6.58)]{GKP}. 
\begin{equation*} 
\gkpSI{n}{2} = (n-1)!\ H_{n-1} = (n-1)!\ (\gamma+\psi(n)) 
\end{equation*} 
The enumeration of \eqref{eqn_Hn_EGF_V1} provides another 
identity for the sequence given by the following equation over the positive 
integer index for $n$ [cf.\ \eqref{eqn_Hn_ident_S1keq2}; 
\sref{subsubSection_HarmonicNumber_Idents}]. 
\begin{equation*} 
\frac{H_n}{n!} = \frac{(-1)^n\ \Gamma(n-\Nc-1)}{(\Nc+1)\ \Gamma(-\Nc-1)\ 
     \Gamma(n+1)^2} \left(\frac{\Nc}{\Nc+1}+\PolyGamma{0}{-\Nc-1}- 
     \PolyGamma{0}{n-\Nc-1}\right) 
\end{equation*} 

\subsubsection{Properties of the Stirling Numbers of the First Kind} 
\label{subsubSection_HarmonicNumber_Idents} 

The first result for the triangle of Stirling numbers of the 
first kind corresponds to the fixed column index $k := 2$ 
\citeseqnum{A000254} and 
is a well-known identity related to the sequence of first-order 
harmonic numbers \cite[\Section 6.3]{GKP}. 
The identity is rephrased by equation \eqref{eqn_Hn_ident_S1keq2} in 
terms of the Stirling triangle result of \eqref{eqn_S1_pfn_spec_case} 
when $\alpha := 1$ and for the same fixed $k := 2$. 
\begin{equation} 
\label{eqn_Hn_ident_S1keq2} 
H_n = \frac{\partial}{\partial \gamma}\left[\widetilde{p}(n)
      \vphantom{\frac{p}{1}}\right]+\widetilde{p}(n) = 
      \frac{(-1)^n\ \Gamma(n-\Nc-1)(\PolyGamma{0}{n-\Nc-1}-
      \PolyGamma{0}{-\Nc}-1)}{\Gamma(n+1)\ \Gamma(-\Nc)} 
\end{equation} 
The original harmonic number identity is equivalent to the property of the 
generalized Bernoulli polynomials discussed in the 
book \booktitle{The Umbral Calculus} \cite[pp.\ 99--100]{UC}. The 
Bernoulli polynomial form of the identity is stated in equation 
\eqref{eqn_UC_gen_BP_Hn_ident} as it appears in the reference. 
\begin{equation} 
\label{eqn_UC_gen_BP_Hn_ident} 
H_n = 1+\cdots+\frac{1}{n} = \frac{(-1)^{n-1}}{(n-1)!}\ 
     B_{n-1}^{(n+1)}(0) 
\end{equation} 
 
The second result is another well-known identity 
that relates the Stirling triangle for the fixed column index $k := 3$ 
\citeseqnum{A000399} \cite{GKP,TAOCP} and the 
first-order and second-order harmonic number sequences 
\citeseqnumIV{A001008}{A002805}{A007406}{A007407}. This classical result is 
restated by equation \eqref{eqn_Hn_ident_V2_S1keq3} and 
is then rephrased by the closed-form identity in terms of the 
polygamma function in \eqref{eqn_Hn_ident_V2_S1keq3_closed_form}. 
\begin{align} 
\label{eqn_Hn_ident_V2_S1keq3} 
\frac{1}{2}\ (H_n^2-H_n^{(2)}) & = \frac{1}{n!} \gkpSI{n+1}{3} = 
     \frac{\partial^2}{\partial \gamma^2}\left[
     \frac{\widetilde{p}(n)}{2}\right]+\frac{\partial}{\partial \gamma}
     \left[\widetilde{p}(n)\vphantom{\frac{p}{1}}\right] \\ 
\label{eqn_Hn_ident_V2_S1keq3_closed_form} 
\frac{1}{2}\ (H_n^2-H_n^{(2)}) & = \frac{(-1)^{n+1} 
     \Gamma(n-\Nc-1)}{2\ \Gamma(n+1)\ 
     \Gamma(-\Nc)}\ \Biggl(\PolyGamma{1}{n-\Nc-1}-\PolyGamma{1}{-\Nc} \\ 
\notag 
   & \phantom{=} +(\PolyGamma{0}{-\Nc}-\PolyGamma{0}{n-\Nc-1}) 
     (2-\PolyGamma{0}{n-\Nc-1}+\PolyGamma{0}{-\Nc})\Biggr) 
\end{align} 
 
Additional identities related to harmonic number sequences that are 
closely related to the form of equations \eqref{eqn_Hn_ident_S1keq2} and 
\eqref{eqn_Hn_ident_V2_S1keq3_closed_form} can be 
constructed in terms of the polygamma functions. 
These additional constructions can be derived from the 
column-based properties of the Stirling triangle as discussed in 
\sref{subsubSection_SymmTransforms_pn_product_function_forms} 
in order to obtain related identities for the other fixed triangle columns
as closed-form functions over the row index $n$. 
For example, when the column index is the fixed 
$k := 4$ \citeseqnum{A000454}, the 
next identity relates the Stirling number triangle and the 
integer-order \keyword{generalized harmonic number} sequences 
\citeseqnumVI{A001008}{A002805}{A007406}{A007407}{A007408}{A007409} 
\cite[\Section 2]{STIRESUMS}. 
\begin{equation*} 
\frac{1}{n!}\ \gkpSI{n+1}{4} = 
     \frac{1}{6}\ \left(H_{n}^3- 3H_{n}H_{n}^{(2)}+2H_{n}^{(3)}\right) 
\end{equation*} 
More generally, consider the function defined recursively by equation 
\eqref{eqn_wnm_rform} where the generalized 
\keyword{$r$-order harmonic numbers} are 
defined as the finite partial sum $H_n^{(r)} := \sum_{k=1}^{n} k^{-r}$. 
\begin{equation} 
\label{eqn_wnm_rform} 
w(n;\ m) := \sum_{k=0}^{m-1} \Pochhammer{1-m}{k}\ H_{n-1}^{(k+1)}\ 
     w(n;\ m-1-k)+\Iverson{m = 0} 
\end{equation} 
 From the definition provided in \eqref{eqn_wnm_rform} it follows that 
the triangle of Stirling numbers of the first kind can be 
expressed in terms of the integer-order harmonic number sequences as in 
equation \eqref{eqn_S1_wnm_rform_eq} where 
$k \in [1,\ n+1] \subseteq \mathbb{N}$ \cite[cf.\ (5)]{STIRESUMS} 
\cite[cf.\ \keyword{hyperharmonic numbers}]{PROPSr-STIRSERIES}. 
\begin{equation} 
\label{eqn_S1_wnm_rform_eq} 
\frac{1}{n!}\ \gkpSI{n+1}{k} = \frac{w(n+1;\ k-1)}{(k-1)!} 
\end{equation} 
The forms obtained from \eqref{eqn_S1_wnm_rform_eq} can then once again be 
rephrased in terms of the results in equations 
\eqref{eqn_S1_pfn_spec_case} and \eqref{eqn_Fa_pGF_Deriv_closedform_ident} 
through the procedure discussed in 
\sref{subsubSection_SymmTransforms_pn_product_function_forms}. 

\subsubsection{Properties of the $r$-Order Harmonic Number Sequences} 

The article by Adamchik provides a concise definition of the 
$r$-order harmonic numbers in terms of higher-order polygamma functions 
through the next equation for positive $r \in \mathbb{N}$ 
\cite[\Section 1; (14)]{STIRESUMS}. 
\begin{align} 
\label{eqn_Adamchik_Hnr_PG_form} 
H_n^{(r)} & = \frac{(-1)^{r-1}}{(r-1)!}\ \left(\PolyGamma{r-1}{n+1}- 
     \PolyGamma{r-1}{1}\right) = 
     \int_{0}^{1} \frac{\Log(t^{-1})^{r-1}}{(r-1)!} \frac{1-t^n}{1-t}\ dt 
\end{align} 
As remarked for the generalized \eqref{Fa_rdef} triangle forms defined by 
equation \eqref{eqn_Fa_pGF_Deriv_closedform_ident} in the previous section, 
the finite expansions in terms of polygamma functions 
that can be obtained for each separate fixed column index of the 
classical Stirling and generalized $\alpha$-factorial triangles also 
serve as variations of the closed-form expansions for cases of the 
$r$-order harmonic number sequences. The following equations 
summarize the forms of the first several non-trivial cases of the 
positive integer-order harmonic number sequences given in terms the 
Stirling triangle coefficients for $\alpha := 1$. 
%% : "working_S1_Hn_sp_case-rorder_Hn_S1_idents.*": 
\begin{align*} 
%H_n & = \frac{1}{n!}\gkpSI{n+1}{2} \\ 
H_n^{(2)} & = \frac{1}{(n!)^2} \gkpSI{n+1}{2}^2-\frac{2}{n!} 
     \gkpSI{n+1}{3} \\ 
H_n^{(3)} & = \frac{1}{(n!)^3} \gkpSI{n+1}{2}^3-\frac{3}{(n!)^2} 
     \gkpSI{n+1}{2} \gkpSI{n+1}{3}+\frac{3}{n!} \gkpSI{n+1}{4} \\ 
H_n^{(4)} & = \frac{1}{(n!)^4} \gkpSI{n+1}{2}^4-\frac{4}{(n!)^3} 
     \gkpSI{n+1}{2}^2 \gkpSI{n+1}{3}+\frac{4}{(n!)^2} \gkpSI{n+1}{2} 
     \gkpSI{n+1}{4}+\frac{2}{(n!)^2} \gkpSI{n+1}{3}^2 \\ 
   & \phantom{=} -\frac{4}{n!} \gkpSI{n+1}{5} 
%H_n^{(5)} & = \frac{1}{(n!)^5} \gkpSI{n+1}{2}^5-\frac{5}{(n!)^4} 
%     \gkpSI{n+1}{2}^3 \gkpSI{n+1}{3}+\frac{5}{(n!)^3} \gkpSI{n+1}{2}^2 
%     \gkpSI{n+1}{4} \\ 
%   & \phantom{=} +\frac{5}{(n!)^3} \gkpSI{n+1}{2} \gkpSI{n+1}{3}^2 
%     +\frac{5}{(n!)^3} \gkpSI{n+1}{2} \gkpSI{n+1}{3}^2-\frac{5}{(n!)^2} 
%     \gkpSI{n+1}{2} \gkpSI{n+1}{5} \\ 
%   & \phantom{=} -\frac{5}{(n!)^2} \gkpSI{n+1}{3} \gkpSI{n+1}{4} 
%     +\frac{5}{n!} \gkpSI{n+1}{6} 
\end{align*} 
In the general case, the closed-forms in terms of the polygamma functions 
that result from computations of these first several sequences of the 
$r$-order harmonic numbers from the identities of 
\sref{subsubSection_SymmTransforms_pn_product_function_forms} are 
significantly more involved than that of equation 
\eqref{eqn_Adamchik_Hnr_PG_form}. That being said, the 
closed-forms are interesting consequences of the identities in 
the previous several sections and the treatment of the constant parameter 
$\Nc$ yields distinct forms that simplify the behavior described by 
the row indexed results somewhat. 

The forms of the $r$-order harmonic sequences enumerated from the 
expansion of these properties can be precisely expressed from the 
recurrence definition of equation \eqref{eqn_wnm_rform}. 
Let the exponential generating function over the single index $m$ for the 
series terms from \eqref{eqn_wnm_rform} be defined as in equation 
\eqref{eqn_WHat_EGF_defn}. 
\begin{equation} 
\label{eqn_WHat_EGF_defn} 
\widehat{W}_{n}(z) := \sum_{m=1}^{\infty} 
     \frac{(-1)^{m}\ w(n+1;\ m)}{m!}\ z^m 
\end{equation} 
It then follows that the generating function for the 
k-order harmonic sequences be defined by the forms of the 
next set of equations. 
\begin{align} 
\label{eqn_Hz_rorder_Hn_OGF_defn} 
H_n(z) & := \sum_{k=1}^{\infty} H_n^{(k)}\ z^k \\ 
\notag 
   & \phantom{:}= \frac{z}{2}\ \frac{\partial}{\partial z}\biggl[ 
     \frac{\widehat{W}_{n}(z)^2}{2\ (2-\widehat{W}_{n}(z)^2)}\biggr]- 
     z\ \frac{\partial}{\partial z}\biggl[\int_{0}^{1}
     \frac{\widehat{W}_{n}(z)}{(1-x^2\ \widehat{W}_{n}(z)^2)} dx\biggr] \\ 
\notag 
   & \phantom{:}= \frac{(4-4\widehat{W}_{n}(z)-4\widehat{W}_{n}(z)^2+ 
     4\widehat{W}_{n}(z)^3+\widehat{W}_{n}(z)^4)\ 
     \widehat{W}_{n}^{\prime}(z)\ z}{ 
     (\widehat{W}_{n}(z)-1) (\widehat{W}_{n}(z)+1) (\widehat{W}_{n}(z)^2-2)^2} 
\end{align} 
The Stirling triangle defines the coefficients of 
\eqref{eqn_WHat_EGF_defn} through the identity \cite[cf.\ (5)]{STIRESUMS} 
\begin{equation*} 
w(n+1;\ m) = \gkpSI{n+1}{m+1}\ \frac{m!}{n!} 
\end{equation*} 
when the index inputs correspond to the 
standard bounds on positive non-zero entries of the triangle. 
However, it should be noted that in general 
the equivalence does not hold in both directions. That is, the 
Stirling triangle coefficients are defined by the enumeration of 
\eqref{eqn_wnm_rform}, but the form of the recurrence in equation 
\eqref{eqn_wnm_rform} is not completely specified by the index cases 
corresponding to the Stirling triangle and 
the well-known classical series for that form. Finding a 
closed-form expression for the exponential generating function in 
\eqref{eqn_WHat_EGF_defn}, and then the resulting identities for the 
generating function in \eqref{eqn_Hz_rorder_Hn_OGF_defn}, remains an 
interesting open problem. 

\subsubsection{Example: Approximations of the Euler-Mascheroni Constant} 

The Euler-Mascheroni constant, $\gamma_{\E}$, \citeseqnum{A001620} 
is defined by the following limit of the difference between the $n^{th}$ 
first-order harmonic number and natural logarithm evaluated at $n$. 
\begin{equation*} 
\lim_{n\rightarrow\infty} \left(H_n-\Log(n)\right) = 
\lim_{n\rightarrow \infty} \gamma_n := \gamma_{\E} \approx 0.5772156649 
\end{equation*} 
The approximate values of the incremental constants, 
denoted $\gamma_n$, may be 
computed from the closed-form given by equation \eqref{eqn_Hn_ident_S1keq2}. 
A summary of computations of the approximations of the constant over 
increasing inputs $n$ is given in \tref{table_EulerGammaConstants}. 
\bigskip  
\begin{numberedtable}[Euler Gamma Constant Approximations] 
\label{EulerGamma_comps_table_pageref} 
\label{table_EulerGammaConstants} 
\begin{center} 
\begin{tabular}{|l|c|l|l|l|l|} 
\hline 
$n$ & $\log_{10}(n)$ & $H_n$ & $\Log(n)$ & $\gamma_n$ & 
$\lvert \gamma_{\E}-\gamma_n \rvert$ \\ \hline 
1 & 0 & 1.000000000 & 0 & 1.000000000 & 0.4227843351 \\ 
10 & 1 & 2.928968254 & 2.302585093 & 0.6263831610 & 0.04916749607 \\ 
100 & 2 & 5.187377518 & 4.605170186 & 0.5822073317 & 0.004991666750 \\ 
1000 & 3 & 7.485470861 & 6.907755279 & 0.5777155816 & 0.0004999166667 \\ 
10000 & 4 & 9.787606036 & 9.210340372 & 0.5772656641 & 0.00004999916667 \\ 
100000 & 5 & 12.09014613 & 11.51292546 & 0.5772206649 & 
     $4.999991667 \times 10^{-6}$ \\ 
1000000 & 6 & 14.39272672 & 13.81551056 & 0.5772161649 & 
     $4.999999167 \times 10^{-7}$ \\ 
\hline 
\end{tabular} 
\end{center} 
\end{numberedtable} 
\bigskip 

\section{The $j$-Factorial Polynomials} 
\label{Section_MF_Fa_polynomials} 

\subsection{Motivation and Background} 
\label{subSection_JFactPolys_Motivations_and_Background} 

The definition of the \keyword{Stirling convolution polynomial} sequence, 
denoted by $\StirlingSigmaPoly{n}{x}$, and corresponding enumerations 
suggest a generalization that results in an extended set of parametrized 
$\alpha$-factorial polynomial sequences, denoted by 
$\FaSigmaPoly{\alpha}{n}{x}$ \cite[\Section 6;\ cf.\ (7.52)]{GKP} 
\cite{CVLPOLYS} \cite[cf.\ \Section 4]{ADELBERGA} 
\cite[cf.\ \Section 4.8]{UC}. 
Additional polynomials related to the classical Stirling number 
triangles considered in the referenced literature include the 
\keyword{C-numbers} \cite[(3.14)]{CS} and 
other combinatorial-based polynomial forms 
defined by the contexts of individual applications 
\cite[\Section 12]{r-STIRLINGNUMS} \cite{STIRLINGPOLYS,MWARDFACTSUMS}. 

The approach to defining the 
generalized polynomials is based on treating the triangle entries in 
equations \eqref{Fa_rdef} and \eqref{Fa2_rdef} as polynomials in the upper 
coefficient index, as is used to define the 
polynomials corresponding to the existing expansions of the classical 
Stirling triangles. The polynomials corresponding to the 
Stirling numbers of the first kind are 
formed by the variant \keyword{Newton series} expansion identity 
in terms of the \quotetext{second-order} Eulerian number triangle 
given by the next equation \citeseqnum{A008517} 
\cite[(6.44) and (6.45); cf.\ (6.43)]{GKP} 
\cite[cf.\ \quotetext{Asymptotics}]{CVLPOLYS}. 
\begin{equation*} 
\gkpSI{x}{x-n} = \sum_{k=0}^{n} \gkpEII{n}{k} \binom{x+k}{2n} 
\end{equation*} 
It happens that considering the coefficient expansions with 
the known factors in the sum removed from the 
expansion of each binomial term results in forms that are particularly 
revealing with respect to the underlying structure of each triangle 
expressed by the generating function enumerating the polynomials. 
The Stirling (convolution) polynomial sequence is defined by the 
enumeration of equation \eqref{eqn_StirlingSigmaPoly_enum_CM_6.50} 
\cite[\Section 6.2]{GKP}. 
\begin{equation} 
\label{eqn_StirlingSigmaPoly_enum_CM_6.50} 
\sum_{n=0}^{\infty} x \StirlingSigmaPoly{n}{x} z^n := 
     \sum_{n=0}^{\infty} \gkpSI{x}{x-n}\ \frac{(x-n-1)!}{(x-1)!}\ z^n = 
     \left(\frac{z e^{z}}{e^{z}-1}\right)^x 
\end{equation} 
The given generating function for the Stirling polynomial sequence 
satisfies the requirements of the more general form of 
\keyword{convolution polynomial} sequences \cite{CVLPOLYS} and 
so may be expressed in terms of the characteristic identities of these forms, 
including the table of convolution properties provided in the 
primary reference on the sequence 
\cite[Table 272 and \Section 6.5]{GKP}. 
Additional properties of the sequence of Stirling polynomials 
in relation to the generalized $\alpha$-factorial polynomial sequences 
are considered in \sref{pageref_subsubSection_Stirling_Polynomial_Forms}. 

\subsection{$j$-Factorial Polynomial Definitions and Generating Functions} 
\label{subSection_MF_Poly_defs_and_enums} 

\subsubsection{Generalized Polynomials of the First Kind} 
\label{subsubSection_FirstKindPolys_Intro_defs} 

The $\alpha$-factorial polynomial sequences are defined by a 
generalization of the Stirling convolution polynomial definition 
\cite[\Section 6.2]{GKP} that results in the equivalent definitions given by 
equation \eqref{Fa_MF_poly_defs1}. 
\begin{align} 
\label{Fa_MF_poly_defs1} 
\FaSigmaPoly{\alpha}{n}{x} := \FcfII{\alpha}{x}{x-n} \frac{(x-n-1)!}{x!} 
     & \quad\iff\quad \FcfII{\alpha}{x}{m} = \frac{x!}{(m-1)!}\ 
       \FaSigmaPoly{\alpha}{x-m}{x} 
\end{align} 
 From equations \eqref{eqn_StirlingSigmaPoly_enum_CM_6.50} and 
\eqref{Fa_MF_poly_defs1} 
it is possible to derive the characteristic generating function forms for these 
parametrized $\alpha$-factorial polynomial sequences from the results of 
\sref{subSection_FiniteDiffIdents} and \sref{Section_Symm_poly_topics} 
giving the generalized coefficients in terms of the 
Stirling triangle (omitted for length). The resulting enumerations are 
denoted by the generating function $S_{\alpha}(x;\ z)$ and are given by the 
equivalent forms in equations \eqref{Fa_MF_poly_GF_enum_exp} and 
\eqref{Fa_MF_poly_GF_enum_trigh}. 
\begin{align} 
\label{Fa_MF_poly_GF_enum_exp} 
S_{\alpha}(x;\ z) & := \sum_{n=0}^{\infty} 
     \FaSigmaPoly{\alpha}{n}{x} z^n = e^{(1-\alpha) z}\ {\left(\frac{\alpha 
     z e^{\alpha z}}{e^{\alpha z}-1}\right)}^x \\ 
\label{Fa_MF_poly_GF_enum_trigh}     
 & \phantom{:}= \left(\cosh((1-\alpha) z)+\sinh((1-\alpha) z)\right) 
   \left(\frac{\alpha z\ (\cosh(\alpha z)+\sinh(\alpha z))}{\cosh(\alpha z)+
   \sinh(\alpha z)-1}\right)^x 
\end{align} 
As with the related generating function for the 
generalized Bernoulli polynomials, it should be 
possible to define the generating functions in terms of a complex-valued 
\keyword{Cauchy contour integral} about the origin as is given in 
Temme's work \cite{BPOLYSOLDANDNEW}, though the form of 
\eqref{Fa_MF_poly_GF_enum_exp} is sufficient to define the 
properties considered in the context of this article. 

The generating function $S_{\alpha}(x;\ z)$ in 
\eqref{Fa_MF_poly_GF_enum_exp} satisfies each of the following 
ordinary differential equation variations in 
\eqref{eqn_diffEQ_S1_form} and \eqref{eqn_diffEQ_SAlpha_form}. 
\begin{align} 
\label{eqn_diffEQ_S1_form} 
S_{1}^{\prime\prime}(x-2;\ -\alpha z)+r_1(z)\ 
     S_{1}^{\prime}(x-2;\ -\alpha z)+r_0(z)\ S_{1}(x-2;\ -\alpha z)- 
     \frac{S_{\alpha}(x;\ z)}{r_2(z)} & = 0 \\ 
\label{eqn_diffEQ_SAlpha_form} 
S_{\alpha}^{\prime\prime}(x-2;\ z)+\frac{p_1(z)}{p_2(z)} 
     S_{\alpha}^{\prime}(x-2;\ z+\frac{p_0(z)}{p_2(z)} 
     S_{\alpha}(x-2;\ z)-\frac{S_{\alpha}(x;\ z)}{p_2(z)} & = 0 
\end{align} 
The coefficient functions $r_i(z)$ in equation \eqref{eqn_diffEQ_S1_form} 
are defined as 
\begin{align*} 
r_0(z) & = \frac{(x-2) (-\alpha (x-2) z+x-1)}{z^2} \\ 
r_1(z) & = \frac{(x-2) (\alpha z-2)}{z} \\ 
r_2(z) & = \frac{z^2 e^{z (\alpha (x-2)+1)}}{(x-2) (x-1)} 
\end{align*} 
and the coefficient functions $p_i(z)$ in equation 
\eqref{eqn_diffEQ_SAlpha_form} are defined as 
\begin{align*} 
p_0(z) & = \frac{e^{\alpha z} (-\alpha (x-2) z+x-1)}{x-1} \\ 
p_1(z) & = \frac{(x-2) z e^{\alpha z} (\alpha z-2) \left(e^{\alpha z} 
     (\alpha z-1)+1\right)}{(x-1) \left(\alpha x z+e^{\alpha z} ((\alpha-1) 
     z-x+2)-3 \alpha z+x+z-2\right)} \\ 
%p_2(z) & = (z^2 e^{\alpha z} (e^{\alpha z} \left(2 x (\alpha z-1)+e^{\alpha z} 
%     \left(x (\alpha z-1)^2-2 \alpha z (\alpha z-2)-3\right)+\alpha z 
%     (\alpha z-4)+6\right) \\ 
%   & \phantom{==} +x-3)) / ((x-1) (e^{2 \alpha z} \left(-2 (\alpha-1) 
%     (x-2) z+(\alpha-1)^2 z^2+x^2-5 x+6\right) \\ 
%   & \phantom{==} +z^2 (\alpha (x-3)+1)^2+e^{\alpha z} (z^2 (\alpha (\alpha 
%     (3 x-8)-2 x+8)-2)-2 (x-2) z (\alpha (x-4) \\ 
%   & \phantom{==} +2)-2 (x-3) (x-2))+2 (x-2) z (\alpha (x-3)+1)+x^2-5 x+6)). 
p_2(z) & = \scriptstyle{
     (z^2 e^{\alpha z} (e^{\alpha z} \left(2 x (\alpha z-1)+e^{\alpha z} 
     \left(x (\alpha z-1)^2-2 \alpha z (\alpha z-2)-3\right)+\alpha z 
     (\alpha z-4)+6\right)+x-3))\ /\ d_2(z)} \\ 
d_2(z) & = \scriptstyle{((x-1) (e^{2 \alpha z} \left(-2 (\alpha-1) 
     (x-2) z+(\alpha-1)^2 z^2+x^2-5 x+6\right) 
     +z^2 (\alpha (x-3)+1)^2 + } \\ 
   & \phantom{=} \quad \scriptstyle{e^{\alpha z} (z^2 (\alpha (\alpha 
     (3 x-8)-2 x+8)-2)-2 (x-2)+z (\alpha (x-4)+2)-2 (x-3) (x-2))+2 (x-2) z 
     (\alpha (x-3)+1)+x^2-5 x+6))}. 
\end{align*} 
A listing of the first several generalized polynomials of the 
first kind is provided in \tref{table_first_kind_polys}. 
\smallskip 
\begin{numberedtable}[The $\mathbf{\alpha}$-Factorial Polynomials of the 
                      First Kind] 
\label{table_first_kind_polys} 
\begin{center} 
\begin{tabular}{|l|l|} 
\hline 
$n$ & $\FaSigmaPoly{\alpha}{n}{x}$ \\ 
\hline 
0 & $1$ \\
1 & $\frac{x \alpha }{2}-\alpha +1$ \\
2 & $\frac{x^2 \alpha ^2}{8}+\frac{1}{24} x \left(12 \alpha -13 \alpha ^2 
     \right)+\frac{1}{2} \left(\alpha ^2-2\alpha +1\right)$ \\
3 & $\frac{x^3 \alpha ^3}{48}+\frac{1}{48} x^2 \left(6 \alpha ^2-7 \alpha ^3 
     \right)+\frac{1}{6} \left(-\alpha^3+3 \alpha ^2-3 \alpha +1\right)+ 
     \frac{1}{24} x \left(7 \alpha ^3-13 \alpha ^2+6 \alpha \right)$ \\
\hline 
\end{tabular} 
\end{center} 
\end{numberedtable} 
\bigskip 

\subsubsection{Generalized Polynomials of the Second Kind} 

In the same manner as the previous section, consider the 
related polynomial definition for the \eqref{Fa2_rdef} triangle given by the 
next equation in \eqref{Fa2_MF_poly_defs1}. 
\begin{equation} 
\label{Fa2_MF_poly_defs1} 
\FaIISigmaPoly{\alpha}{n}{x} := \FIIcfII{\alpha}{x}{x-n} 
     \frac{(x-n-1)!}{x!} \quad\iff\quad \FIIcfII{\alpha}{x}{m} = 
     \frac{x!}{(m-1)!}\FaIISigmaPoly{\alpha}{x-m}{x} 
\end{equation} 
A variant form of the 
polynomials defined by \eqref{Fa2_MF_poly_defs1} that is inspired by the 
Stirling polynomial definitions offered by Adelberg 
\cite[(4.6)]{ADELBERGA} \cite{STIRLINGPOLYS} is defined and 
enumerated by the interesting form in equation 
\eqref{eqn_Fa2_fPoly_def_and_enum} \cite[cf.\ (6.48) and (6.53)]{GKP}. 
\begin{equation} 
\label{eqn_Fa2_fPoly_def_and_enum} 
f_n^{\alpha}(x) = (n+x)\ \FaIISigmaPoly{\alpha}{n}{n+x} \implies  
     \sum_{n=0}^{\infty} f_n^{\alpha}(x)\ z^n = e^{z}\ \left( 
          \frac{\alpha z}{e^{\alpha z}-1}\right)^{1-x} 
\end{equation} 
A listing of the first several examples of the 
\keyword{variant polynomials of the second kind} is provided in 
\tref{table_variant_second_kind_polys}. 

\bigskip 
\begin{numberedtable}[The Variant $\mathbf{\alpha}$-Factorial 
                      Polynomials of the Second Kind] 
\label{table_variant_second_kind_polys} 
\begin{center} 
\begin{tabular}{|l|l|} 
\hline 
$n$ & $f_n^{\alpha}(x)$ \\ 
\hline 
0 & $1$ \\
1 & $\frac{2-\alpha }{2}+\frac{x \alpha }{2}$ \\
2 & $\frac{x^2 \alpha ^2}{8}+\frac{1}{24} x \left(12 \alpha -5 \alpha ^2\right) 
     +\frac{1}{12} \left(\alpha ^2-6\alpha +6\right)$ \\
3 & $\frac{x^3 \alpha ^3}{48}+\frac{1}{12} \left(\alpha ^2-3 \alpha +2\right) 
     +\frac{1}{24} x^2 \left(3 \alpha^2-\alpha ^3\right)+\frac{1}{48} x 
     \left(\alpha ^3-10 \alpha ^2+12 \alpha \right)$ \\ 
%4 & $\frac{x^4 \alpha ^4}{384}+\frac{1}{720} \left(-\alpha ^4+30 \alpha ^2-60 
%     \alpha +30\right)+\frac{1}{192}x^3 \left(4 \alpha ^3-\alpha ^4\right)+ 
%     \frac{x^2 \left(\alpha ^4-48 \alpha ^3+72 \alpha^2\right)}{1152}+ 
%     \frac{1}{960} x \left(3 \alpha ^4+20 \alpha ^3-100 \alpha ^2+80 \alpha 
%     \right)$ \\ 
\hline 
\end{tabular} 
\end{center} 
\end{numberedtable} 
\bigskip 

\subsection{$j$-Factorial Polynomial Identities} 
\label{subSection_MF_Poly_props_and_ident_forms} 

\subsubsection{Initial Polynomial Properties} 
\label{subsubSection_JFactPoly_Initial_Properties} 

As a first observation, the generating function defining the form of the 
polynomials can be shown through the methods of Graham and Knuth 
\cite{GKP,CVLPOLYS} to produce polynomials of finite-degree such that 
$\deg\{\FaSigmaPoly{\alpha}{n}{x}\} = n$. This property is characteristic of 
polynomial sequences with generating functions in the general form of 
equations \eqref{Fa_MF_poly_GF_enum_exp} and 
\eqref{eqn_Fa2_fPoly_def_and_enum} involving a generating series raised to the 
power of the polynomial variable \cite{UC}. 

A simple pair of identities relating the polynomial parameter $\alpha$ 
multiplied by a scalar is formulated for each of the polynomial variations 
by the following equations. 
A notable special case of the first triangle polynomial identity results 
when $c = -1$. 
\begin{align*} 
\FaSigmaPoly{c \alpha}{n}{x} & = \sum_{k=0}^{n} 
     \frac{c^k (1-c)^{n-k}}{(n-k)!}\ \FaSigmaPoly{\alpha}{k}{x} \\ 
\FaIISigmaPoly{c \alpha}{n}{n+x} & = \sum_{k=0}^n \frac{(c-1)^{n-k}}{c^n\ 
     (n-k)!} \frac{(k+x)}{(n+x)}\ \FaIISigmaPoly{c^2 \alpha}{k}{k+x} 
\end{align*} 

The recursive identities in the next equations 
are immediate consequences of the respective definitions 
\eqref{Fa_MF_poly_defs1} and \eqref{Fa2_MF_poly_defs1}. 
These identities provide the most basic recursive properties for the 
sequences that involves a finite shift in each polynomial input 
\cite[cf.\ \Section 3.1]{UC}. 
\begin{align} 
\label{Fa_poly_internal_idents1}
(x+1)\ \FaSigmaPoly{\alpha}{n}{x+1} & = 
     (x-n)\FaSigmaPoly{\alpha}{n}{x}+
     (\alpha x+1-\alpha)\FaSigmaPoly{\alpha}{n-1}{x} \\ 
\notag 
x\ \FaIISigmaPoly{\alpha}{n}{x+1} & = (x-n)\ 
     \FaIISigmaPoly{\alpha}{n}{x}+\left(\alpha x-\alpha n+1\right)\ 
     \FaIISigmaPoly{\alpha}{n-1}{x} 
\end{align} 
The case of equation \eqref{Fa_poly_internal_idents1} when $\alpha := 1$ 
corresponds to the recursive definition of the 
original Stirling polynomial sequence \cite[p.\ 311;\ \Section 6]{GKP}. 

Finally, the polynomials of the first and second kinds are related through the 
next pair of conversion formulas for a fixed parameter $\alpha$. 
The given conversion identities follow from the discrete convolution of 
terms in the series expansions for the generating functions of equations 
\eqref{Fa_MF_poly_GF_enum_exp} and \eqref{eqn_Fa2_fPoly_def_and_enum} 
\cite[cf.\ (6.33) and (6.48)]{GKP} 
[cf.\ \sref{subSection_GenStirlingNums_forms_and_diffs};\ 
\sref{pageref_subsubSection_Stirling_Polynomial_Forms}]. 
\begin{align*} 
%% : "Fa2_poly_extra_idents.*": 
 & \FaIISigmaPoly{\alpha}{n}{n+x} = \frac{(1-x)}{(n+x)} \sum_{k=0}^n 
     \frac{(\alpha x)^k}{k!}\ \FaSigmaPoly{\alpha}{n-k}{1-x} \\ 
 & \FaSigmaPoly{\alpha}{n}{x} = \sum_{k=0}^n \sum_{j=0}^{n-k} 
     \frac{(\alpha(x-1)-1)^{n-k-j}}{x\ (n-k-j)!} (j+2-2x) (k+x)\ 
     \FaIISigmaPoly{\alpha}{j}{j+2-2x} \FaIISigmaPoly{\alpha}{k}{k+x} 
\end{align*} 
Given the coefficient definitions for each polynomial variation from 
equations \eqref{Fa_MF_poly_defs1} and \eqref{Fa2_MF_poly_defs1}, 
these conversion identities also provide new identities 
relating the triangles in \eqref{Fa_rdef} and \eqref{Fa2_rdef} and extend the 
coefficient properties discussed in \sref{subSection_FiniteDiffIdents}. 
The forms that result from the special case when $\alpha := 1$ offer an 
unusual pair of identities relating the classical Stirling coefficients 
as well as the classical triangles to the \quotetext{second-order} 
Eulerian number triangle through closed-form double and quadruple sums 
\cite[cf.\ Table 265;\ (6.43) and (6.44)]{GKP}. 

\subsubsection{Polynomial Results Based on Properties of the First Triangle} 
\label{subsubSection_FirstPolyResults_v1} 

In the following discussion several forms of the 
$\FaSigmaPoly{\alpha}{n}{x}$ polynomials are 
derived from identities and resulting generating functions 
based on the properties of \sref{subSection_init_Fa_symm_idents} 
[cf. \sref{subsubsection_BPolyEnums_of_FirstTriangle}]. 

First, by applying \eqref{eqn_pnj_form_pn0_coeff_of_ni} to the 
identity in equation \eqref{eqn_Fa_pnj_sum_form}, the resulting form of the 
\eqref{Fa_rdef} triangle coefficients is given by the following equation in 
\eqref{eqn_RHS_px_Fa_poly_ident}. 
\begin{align} 
\label{eqn_RHS_px_Fa_poly_ident} 
\FcfII{\alpha}{x}{x-n}\ \frac{(x-n-1)!}{x!} & = 
     \sum_{k=0}^{x-2} \sum_{i=0}^{n-1} \frac{(\alpha \Xc)^{i}\ (-1)^{n-1-i}\ 
     \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1}\ (2\alpha-2)^{k-x}}{ 
     \Xc\ (2\alpha-2)^{i-n-1}\ i!\ (n-1-i+k+2-x)!} \\ 
\notag 
   & \phantom{=} + \sum_{k=0}^{x-2} \sum_{i=0}^{n} \frac{
     (\alpha \Xc)^{i}\ (x-n-1)\ (-1)^{n-i}\ \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1}\ 
     (2\alpha-2)^{k-x}}{\Xc (\Xc-1)\ (2\alpha-2)^{i-n-2}\ i!\ 
     (n-i+k+2-x)!} 
\end{align} 
Let the functions $S_i(x;\ z)$ be defined as follows: 
\begin{align*} 
S_1(x;\ z) & := \sum_{k=0}^{x-2} \sum_{n=0}^{\infty} \frac{ 
     -(2\alpha-2)^{k+2-x}\ \alpha^{n}\ \Xc^{n-1}\ 
     \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1}}{(\Xc-1)\ \Gamma(n+1)\ 
     \Gamma(k+2-x)}\ z^n \\ 
S_2(x;\ z) & := \sum_{k=0}^{x-2} \sum_{n=0}^{\infty} \frac{ 
     (-1)^{x-k}\ (2+(\Xc-2)\ \alpha)^{n}\ \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1} 
     }{\Xc\ (\Xc-2)\ (1+2k-n+(k-n)(\Xc-2)\ \alpha)^{-1}\ \Gamma(n+2)}\ 
     z^{n+k+3-x} \\ 
S_3(x;\ z) & := \sum_{k=0}^{x-2} \sum_{n=0}^{\infty} \frac{ 
     (-1)^{x+1-k} (2+(\Xc-2) \alpha)^{n-1} 
     \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1}}{\Xc (\Xc-1) 
     (2+2k-n+(k-n+1)(\Xc-2) \alpha)^{-1} \Gamma(n+1)}\ z^{n+k+2-x}. 
\end{align*} 
 From each side of equation \eqref{eqn_RHS_px_Fa_poly_ident}, and by the 
application of \eqref{Fa_MF_poly_defs1}, it can be shown that 
\begin{align*} 
\sum_{n=0}^{\infty} \FaSigmaPoly{\alpha}{n}{x} z^n & = \phantom{} 
     \sum_{k=0}^{x-2} \frac{- e^{\alpha \Xc\ z} (2\alpha-2)^{k-x+2}\ 
     \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1}}{\Xc\ (\Xc-1)\ \Gamma (k-x+2)} \\ 
   & \phantom{=} +\sum_{k=0}^{x-2} \frac{e^{(2+(\Xc-2)\alpha) z}\ 
     (\alpha-1)^{k-x}\ ((1+(\Xc-2)\alpha)\ z-(k+1))\ 
     \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1}}{\Xc\ (\Xc-1)\ (z-\alpha z)^{k-x}\ 
     z^2} \\ 
   & \phantom{=} +\sum_{k=0}^{x-2} \frac{-e^{(2+(\Xc-2)\alpha) z}\ 
     (2\alpha-2)^{k-x+2}\ E_{x-1-k}((2-2\alpha) z)\ 
     \FaSigmaPoly{\alpha}{x-2-k}{\Xc-1}}{\Xc\ (\Xc-1)\ 
     ((1+(\Xc-2)\alpha)\ z-(k+1))^{-1}\ \Gamma(k+2-x)}
\end{align*} 
where $E_k(x)$ denotes the exponential integral function. 
It follows from the given expressions that the next polynomial identity 
holds in terms of the functions $S_i(x;\ z)$. 
\begin{equation*} 
\FaSigmaPoly{\alpha}{n}{x} = \frac{(-1)^{n+1}\ x\ (x+1)}{(x-n)}\ 
     [z^{x+1-n}]\left(S_1(x+1;\ z) z^x+S_2(x+1;\ z) z^x 
     +S_3(x+1;\ z) z^x\right) 
\end{equation*} 

Next, for $x \in [2,\ \infty) \subseteq \Nset$ and 
$n \in [0,\ x-2] \subseteq \Nset$, let the 
coefficients $c(x;\ n) := [t^x z^n] C(t;\ z)$ be defined by the 
following equation. 
\begin{equation*} 
c(x;\ n) = \frac{(x-1)\Iverson{n = 0}}{\Xc (\Xc-1)} 
     +\frac{-\left(2-2x+n+(1+n-x)(\Xc-2) \alpha\right)}{\Xc (\Xc-1) 
     (2+(\Xc-2) \alpha)^{1-n}\ \Gamma(n+1)} \Iverson{n \geq 1} 
\end{equation*} 
Starting from the form of equation \eqref{eqn_RHS_px_Fa_poly_ident}, let the 
right hand side bivariate \quotetext{double} generating function 
$R$ be defined from the coefficient results of the following equations. 
\begin{align*} 
R(t;\ z) & := \sum_{x=2}^{\infty} \sum_{n=0}^{\infty} 
     \FcfII{\alpha}{x}{x-n} \frac{(x-n-1)!}{x!} = 
     C(t;\ z) \ e^{(\alpha-1) tz} \left(\frac{-\alpha tz e^{-\alpha tz}}{ 
     e^{-\alpha tz}-1}\right)^{\Xc-1} \\ 
[z^n] R(t;\ z) & \phantom{:}= \sum_{k=0}^{n} 
     \FaSigmaPoly{\alpha}{n-k}{\Xc-1} \Xc\ (-1)^{n-k} \cdot 
     \left(\sum_{i=0}^{\infty} c(i;\ k)\ t^{i+n-k}\right) \\ 
[t^x z^n] R(t;\ z) & \phantom{:}= \sum_{k=0}^{n} 
     \FaSigmaPoly{\alpha}{n-k}{\Xc-1} \Xc\ (-1)^{n-k}\ c(x-n+k;\ k) 
\end{align*} 
The last of the coefficient identities for the 
function $R(t;\ z)$ provides the polynomial 
identity given by equation \eqref{eqn_RHS_LHS_FaSPoly_ident}. 
\begin{align} 
\label{eqn_RHS_LHS_FaSPoly_ident} 
\FaSigmaPoly{\alpha}{n}{x} & = \frac{(-1)^{n}\ (x-1-n)}{x-1}\ 
     \FaSigmaPoly{\alpha}{n}{x-1} \\ 
\notag 
   & \phantom{=} +\sum_{k=1}^{n} \frac{(-1)^{n-k} 
     (k+2x-2n-2-(n-x+1)(x-2) \alpha)}{(x-1) (\alpha x-2 \alpha+2)^{1-k}\ 
     \Gamma(k+1)}\ \FaSigmaPoly{\alpha}{n-k}{x-1} 
\end{align} 
 From the recurrence in equation \eqref{Fa_poly_internal_idents1} and the 
previous identities for the coefficients of the 
last generating function $R$, the 
next equations for the component functions $R(z)$ and $L(z)$ give the 
following ordinary differential equations in the 
respective polynomial generating functions of equation 
\eqref{Fa_MF_poly_GF_enum_exp}. 
\begin{align*} 
R(z) & := \sum_{n=0}^{\infty} \left(\FaSigmaPoly{\alpha}{n}{\Xc-1} 
     (-1)^{n}+\sum_{k=1}^{n} \FaSigmaPoly{\alpha}{n-k}{\Xc-1} \Xc 
     (-1)^{n-k}\ c(\Xc-n+k;\ k)\right) z^n \\ 
   & \phantom{:}= \frac{e^{(2+(X-2) \alpha) z} z}{\Xc-1} 
     S_{\alpha}^{\prime}(x-1;\ -z)-\frac{((\alpha z-1)\ \Xc-2 \alpha z+z+1) 
     }{(\Xc-1)\ e^{-(2+(\Xc-2) \alpha) z}}\ S_{\alpha}(x-1;\ -z) \\ 
L(z) & := \sum_{n=0}^{\infty} \left(\frac{(x-1-n)}{x}\ 
     \FaSigmaPoly{\alpha}{n}{x-1}+\frac{(\alpha x+1-2\alpha)}{x}\ 
     \FaSigmaPoly{\alpha}{n-1}{x-1}\right) z^n \\ 
   & \phantom{:}= -\frac{z}{x}\ S_{\alpha}^{\prime}(x-1;\ z)+ 
     \frac{((\alpha x-2 \alpha+1) z+x-1)}{x}\ S_{\alpha}(x-1;\ z) 
\end{align*} 
The full form of the ordinary differential equation satisfied by the 
polynomial sequence generating function, $S_{\alpha}(x-1;\ \pm z)$, then 
results from the component functions as the difference $L(z) - R(z) = 0$. 

\subsubsection{Generalized Bernoulli Polynomial Properties} 
\label{subsubSection_GenBernoulliPoly_ident_forms} 

The $\alpha$-factorial polynomials polynomials of the first kind are 
closely related to the generalized Bernoulli polynomials, 
denoted by $B_n^{(\alpha)}(x)$, including the subset of 
N\"{o}rlund polynomials, denoted $B_n^{(\alpha)} = B_n^{(\alpha)}(0)$, 
through the generating function enumerating Bernoulli polynomials in 
equation \eqref{eqn_GenBPoly_enum}. The generalized Bernoulli 
polynomials are studied extensively in the works of Adelberg and many others 
\cite{ADELBERGA,NORLUNDARITHPROPS,CARLITZC,NORLUNDPOLYPROPS,NORLUNDN} 
\cite[\Section 2.2]{UC}. 
\begin{equation} 
\label{eqn_GenBPoly_enum} 
\sum_{n=0}^{\infty} \frac{B_n^{(\alpha)}(x)}{n!}\ z^n = 
     e^{x z}\ \left(\frac{z}{e^{z}-1}\right)^{\alpha} 
\end{equation} 
One possible application of the identities relating the \eqref{Fa_rdef} and 
\eqref{Fa2_rdef} triangles to the higher-order Bernoulli number 
sequences is to generalize the polynomial forms, and then the 
corresponding triangle coefficients through equations 
\eqref{Fa_MF_poly_defs1} and \eqref{Fa2_MF_poly_defs1}, to 
non-integral inputs for $\alpha$ and $x$ 
[cf.\ \sref{subSection_FiniteDiffIdents};\ $t$ in 
\eqref{eqn_Szt_form_for_FaPolys2}]. 
Related generalizations of the classical Stirling numbers of the 
first kind are given in the work of Adamchik and others 
\cite{STIRESUMS,CARLITZC,ARXGENSTIRLING,NORLUNDPOLYEFORMS} 
\cite[(67) and (68)]{r-STIRLINGNUMS} 
\cite[\Section 3]{GENBPEPIDENTS} \cite[\Section3]{BPOLYSOLDANDNEW}. 

Note that while the generalized Bernoulli polynomial forms are 
interesting and suggest a number of external implications for the 
$\alpha$-factorial polynomial sequences based 
on the vast literature on these forms, this particular treatment is not a 
significant influence on the exposition of the article. 
Rather, as with the treatment of the individual expansion coefficients, the 
interesting and important properties of the polynomials are discussed with 
emphasis on the forms provided for the generalized factorial 
function expansions defined in terms of these polynomials and 
more importantly on the 
motivating applications based on the forms as discussed in 
\sref{Section_Symm_poly_topics} and later in 
\sref{Section_JFact_Poly_Applications}. 

Identities for the generalized $\alpha$-factorial polynomials of the 
first kind in equation \eqref{Fa_MF_poly_defs1} that are given in terms of the 
N\"{o}rlund polynomials and the higher-order (generalized) Bernoulli numbers 
include the following forms: 
\begin{align} 
\label{Fa_in_terms_of_other_forms_ident_listings} 
\FaSigmaPoly{\alpha}{n}{x} & = \frac{1}{n!}\ \sum_{k=0}^{n} 
     \binom{n}{k}\ (1+(x-1)\ \alpha)^{k}\ \alpha^{n-k}\ B_{n-k}^{(x)} \\ 
\notag 
  & = \frac{1}{n!}\ \sum_{k=0}^{n} \binom{n}{k}\ 
    (1+(x-2)\ \alpha)^{k}\ \alpha^{n-k}\ B_{n-k}^{(x)}(1) \\ 
\notag 
\FaSigmaPoly{\alpha}{n}{x} & = \alpha^n\ \sum_{k=0}^n 
     \frac{B_k^{(x)}(\frac{1}{\alpha}-1)\ x^{n-k}}{k!\ (n-k)!} \\ 
\notag 
\FaSigmaPoly{\alpha}{n}{x} & = \sum_{k=0}^{n} 
     \frac{\alpha^k\ (1-\alpha)^{n-k}}{k!\ (n-k)!}\ B_k^{(x)}(x) \\ 
\notag 
   & = \sum_{k=0}^{n} \frac{\alpha^k\ (1-\alpha)^{n-k}\ (1+\alpha z)^{n-k} 
     }{k!\ (n-k)!}\ B_k^{(x)}(\alpha x). 
\end{align} 
The generalized $\alpha$-factorial polynomials of the second kind, 
$\FaIISigmaPoly{\alpha}{n}{x}$, are expressed similarly in 
closed-form through the N\"{o}rlund polynomials by equation 
\eqref{Fa2_poly_closedform_in_NorlundB_sum} and in terms of the 
generalized Bernoulli polynomials by equation 
\eqref{Fa2_poly_closedform_in_gen_NorlundB_BP_sum}. 
\begin{align} 
\label{Fa2_poly_closedform_in_NorlundB_sum} 
\FaIISigmaPoly{\alpha}{n}{x} & = \sum_{k=0}^{n} \frac{
     B_k^{(1-x+n)} \alpha^k}{k!\ (n-k)!} \\ 
\label{Fa2_poly_closedform_in_gen_NorlundB_BP_sum} 
\FaIISigmaPoly{\alpha}{n}{x} & = \sum_{k=0}^{n} 
     \frac{\alpha^k\ (1-\alpha)^{n-k}}{k!\ (n-k)!}\ B_k^{(1-x+n)}(1) 
\end{align} 
Each of these initial identities follows directly from the generating 
functions for the respective polynomial sequences compared to 
variations of the parameter forms in \eqref{eqn_GenBPoly_enum}. 

The sequences of Bernoulli polynomials, denoted by 
$B_n(x) := B_n^{(1)}(x)$, \keyword{Euler polynomials}, 
$E_n(x) := E_n^{(1)}(x)$, and \keyword{Hermite polynomials}, 
$H_n^{(v)}(x)$, each satisfying the treatment of an \keyword{Appell sequence} 
given in \booktitle{The Umbral Calculus} reference, have an 
interesting property termed as the 
\keyword{multiplication theorem} \cite[Thm.\ 2.5.10; \Section 2.2]{UC}. 
A similar property for the $\alpha$-factorial polynomial form 
is formulated by equation \eqref{eqn_FaSPoly_multiplication_theorem}. 
\begin{equation} 
\label{eqn_FaSPoly_multiplication_theorem} 
\FaSigmaPoly{\alpha}{n}{mx} = \sum_{i+j+k= n} \left(\sum_{r=0}^{m-2} 
     \frac{\alpha^{j+k}\ (m-1)^{k-1}}{j!\ k!}\ \FaSigmaPoly{\alpha}{i}{x} 
     B_{j}^{(mx-x-1)}\ B_k\left(x+\frac{r}{m-1}\right)\right) 
\end{equation} 
The multiplication theorem also provides the $\alpha$-factorial function 
identity given by the next equation where the input $s$ is considered as a 
formal parameter in the distinct polynomial expansions that result from the 
respective triangle and polynomial forms over arbitrary index input 
\cite[cf.\ (5.83)]{GKP}. 
\begin{equation*} 
\frac{(s-1)!_{(\alpha)}}{(mx)!} = \sum_{k=1}^{mx} 
     \FcfII{\alpha}{mx}{k}\ \frac{(-1)^{mx+k}\ s^{k-1}}{(mx)!} = 
     \sum_{n=1}^{mx} \frac{(-1)^{mx+n}\ s^{n-1}}{(n-1)!}\ 
     \FaSigmaPoly{\alpha}{mx-n}{mx} 
\end{equation*} 

Applications involving the $\alpha$-factorial polynomials of the first kind 
often involve identities for the index diagonals defined as a fixed 
offset over both polynomial inputs. 
Given this observation it then becomes necessary to 
establish properties for the cases of these diagonal index forms. 
For well-defined indices $n \in \mathbb{N}$ and 
$k \in [0,\ x] \subseteq \mathbb{N}$, define the form of 
$\FaSigmaPoly{\alpha}{x-k}{x}$ by the following equations where 
$\Gamma(k;\ z)$ denotes the incomplete gamma function 
[cf.\ \sref{pageref_subsubSection_Stirling_Polynomial_Forms}]. 
\begin{align*} 
\FaSigmaPoly{\alpha}{x-k}{x} & = \sum_{j=0}^{x} \binom{x}{j} 
     \frac{(1-\alpha+\alpha x)^{j} B_{x-j}^{(x)}}{\alpha^{j-x}\ x!} 
     \Iverson{k = 0} + \sum_{j=0}^{x-k} \gkpSI{x-1}{j+k-1} 
     \frac{\alpha^{x-k}\ (j+k-1)!}{\alpha^{j}\ (x-1)!\ j!} 
     \Iverson{k \geq 1} \\ 
   & = [z^x] \left(\frac{(-1)^{k-1}\ \alpha^{x-k}\ z\ \Log(1-z)^{k-1}}{ 
     (1-z)^{\frac{1}{\alpha}}\ \Gamma(x-k+1)}\ 
     \Gamma(x-k+1;\ -\Log(1-z)\ \alpha^{-1})\right) 
\end{align*} 
The second diagonal identity results by applying the classical result for the 
exponential generating function enumerating the Stirling numbers of the 
first kind in equation \eqref{eqn_CMathTable351GF_7.50} \cite[Table 351]{GKP} 
in the specific form of the equation 
\begin{equation*} 
\sum_{n=1}^{\infty} \gkpSI{n-1}{m}\ \frac{m!}{(n-1)!}\ z^n = 
     (-1)^{m}\ z \Log(1-z)^{m} 
\end{equation*} 
to the first diagonal index property. 

The recursive form for the derivative of the polynomial 
$\FaSigmaPoly{\alpha}{n}{x}$ with respect to the variable $x$ is derived 
below where $\langle B_n \rangle$ denotes the classical sequence of 
\keyword{Bernoulli numbers} \citeseqnumII{A000367}{A002445}. 
\begin{align*} 
\frac{d}{dx}\Bigl[\FaSigmaPoly{\alpha}{n}{x}-\FaSigmaPoly{\alpha}{n-1}{x} 
     \Bigr] & = \frac{d}{dx}\Biggl[\alpha^n \StirlingSigmaPoly{n}{x}+
     \sum_{k=0}^{n-1} \frac{(\alpha+n-1-k)\ \alpha^k\ (1-\alpha)^{n-1-k}}{ 
     (n-k)!}\ \StirlingSigmaPoly{k}{x}\Biggr] \\ 
   & = [z^n] \left(e^{(1-\alpha)z} \left(\frac{
     \alpha z e^{\alpha z}}{e^{\alpha z}-1} \right)^x (1-z) 
     \Log\left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)\right) \\ 
   & = \frac{\alpha}{2}\ \FaSigmaPoly{\alpha}{n-1}{x}+ 
     \sum_{k=2}^{n} {\frac{\alpha^{k-1}\ \FaSigmaPoly{\alpha}{n-k}{x}}{ 
     k(k-1)\ \Gamma(k+1)}\ \Bigl(k^2 B_{k-1}-\alpha(k-1) B_k\Bigr)} 
\end{align*} 
These identities provide an analog to standard properties often cited for 
other well-known polynomial sequences. In particular, the identities 
parallel the form of a characteristic property of Appell sequences 
satisfying $s_n^{\prime}(x) := n s_{n-1}(x)$ over all $n \in \mathbb{N}$. 

Exponential generating functions and related coefficient identities for the 
polynomials of the first kind are given by 
equations \eqref{Fa_SPoly_EGF_coeff_terms1}, \eqref{Fa_SPoly_EGF_eqn2}, 
and \eqref{Fa_SPoly_EGF_eqn2_V2}, where $\Gamma(k;\ z)$ denotes the 
incomplete gamma function \cite[cf.\ \Section 9]{ARXONASYM} and 
$I_k(z)$ is the modified Bessel function of the first kind 
\cite[\Section 5.5]{GKP} \cite[cf.\ \Section 1.7]{UC}. 
\begin{align} 
\label{Fa_SPoly_EGF_coeff_terms1} 
\frac{\FaSigmaPoly{\alpha}{n}{x}}{n!}\phantom{z^n} & = [t^n] \left( 
     e^{(1+\alpha(x-1)) t} \left(\frac{\alpha t}{e^{\alpha t}-1}\right)^{x} 
     \frac{\Gamma(n+1;\ (1+\alpha(x-1)\ t))}{\Gamma(n+1)^2}\right) \\ 
\label{Fa_SPoly_EGF_eqn2} 
\sum_{n=0}^{\infty} \frac{\FaSigmaPoly{\alpha}{n}{x}}{n!} z^n & = 
     \sum_{k=0}^{\infty} \frac{\alpha^k\ z^{k/2}}{\Gamma(k+1)}\ 
     I_{-k}(2\sqrt{z})\ B_k^{(x)}(x-1) \\ 
\label{Fa_SPoly_EGF_eqn2_V2} 
\sum_{n=0}^{\infty} \frac{\FaSigmaPoly{\alpha}{n}{x}}{n!} z^n & = 
     \sum_{k=0}^{\infty} \sum_{j=0}^{k} \frac{(-\alpha\sqrt{z})^k\ 
     (-1)^j}{(k-j)!}\ I_{-k}(2\sqrt{z})\ \StirlingSigmaPoly{j}{x} 
\end{align} 
The form of the generalized Bernoulli polynomial in \eqref{Fa_SPoly_EGF_eqn2} 
is similar to that considered by equation 
\eqref{eqn_SAGenApprox_BP_SpecCaseExpansion} in 
\sref{subsubSection_SAGen_ResultDeriv} and leads to the 
special case form of equation \eqref{Fa_SPoly_EGF_eqn2_V2}. 

\subsubsection{Bernoulli Polynomial Enumerations Based on 
               Properties of the First Triangle} 
\label{subsubsection_BPolyEnums_of_FirstTriangle} 

It is possible to derive an alternate form for the 
$\alpha$-factorial polynomial two-variable \quotetext{\keyword{super}} 
generating series coefficients from the 
identities considered in \sref{subSection_init_Fa_symm_idents}. 
The derivation begins in a similar manner to that of 
\sref{subsubSection_FirstPolyResults_v1} and yields a resulting 
generating function involving coefficient terms in the 
N\"{o}rlund polynomials and the exponential integral function, $E_n(z)$. 

%% : "FwzX_GF_from_fa_invs.*": 
To begin with, observe that for integer 
$x \in [2,\ \infty) \subseteq \mathbb{N}$ and 
$n \in [0,\ x-2] \subseteq \mathbb{N}$ the following triangle identity holds. 
\begin{align*} 
\FcfII{\alpha}{x}{x-n}\ \frac{(x-n-1)!}{x!} & = \sum_{i=0}^{x-2} 
     \left(\frac{(\alpha \Xc)^{n+1+i-x}}{\Xc\ (n+1+i-x)!}+ 
     \frac{(\alpha \Xc)^{n+2+i-x}\ (x-n-1)}{\Xc\ (n+2+i-x)!}\right) \times \\ 
   & \phantom{= \sum_{i=0}^{x-2}} \times 
     \left(\sum_{k=0}^{x-2-i} (-1)^{x-2-i-x} 
     \FaSigmaPoly{\alpha}{x-2-i-k}{\Xc-1} \frac{(2-2\alpha)^{k}}{k!}\right) 
\end{align*} 
It follows from the last identity that the polynomial result of the 
next equation also holds. 
\begin{align} 
\label{eqn_FaSPolyGF_FwzX_V1} 
\sum_{m=2}^{\infty} \sum_{n=0}^{\infty} \FaSigmaPoly{\alpha}{n}{x} w^x z^n 
    & = \left(\frac{w^2 e^{(w-\alpha w+\alpha \Xc) z} 
      (1+(\alpha \Xc-1) (w-1) z)}{(\Xc-1)\ (w-1)^2} \right) 
      \left(\frac{\alpha wz}{e^{\alpha wz}-1} \right)^{\Xc-1} 
\end{align} 
The generating function from equation \eqref{eqn_FaSPolyGF_FwzX_V1} is 
expressed by the product of a pair of series where the coefficients of the 
first series expansion may be expressed as follows: 
\begin{align*} 
[w^m] & \left(\frac{w^2\ e^{(w-\alpha w+\alpha \Xc) z} 
     (1+(\alpha \Xc-1)(w-1) z)}{(\Xc-1)\ (w-1)^2}\right) \\ 
   & = 
     \begin{cases} 
          \frac{e^{\alpha \Xc z} (1+z-\alpha \Xc z)}{\Xc-1}, & 
          \text{if $m = 2$; } \\ 
          \frac{(z-\alpha z)^m e^{(1+(\Xc-1)\ \alpha) z} 
          \left((m-2) E_{3-m}(z-\alpha z) (\alpha z-\alpha \Xc z+m-1)+ 
          e^{(\alpha-1) z} (z-\alpha \Xc z+m-1)\right)}{(\Xc-1) 
          (\alpha-1)^2 z^2 \Gamma (m-1)}, & 
          \text{if $m \geq 3$; } \\ 
          0, & \text{otherwise.} 
     \end{cases} 
\end{align*} 
The generating function coefficients on the diagonal of the 
second series expansion are given by the N\"{o}rlund polynomial forms in the 
next equation. 
\begin{equation*} 
\left(\frac{\alpha w z}{e^{\alpha w z}-1}\right)^{\Xc-1} = 
     \sum_{k=0}^{\infty} \frac{\alpha^k\ B_k^{(\Xc-1)}}{k!}\ w^k z^k. 
\end{equation*} 
The full generating series expansion for the product in equation 
\eqref{eqn_FaSPolyGF_FwzX_V1} is then 
defined by the coefficients of the formal variable $z$ formed by 
a discrete convolution of the series terms provided from each of the 
separate generating function expansions. 

\subsubsection{Properties Related to the Stirling Polynomial Sequence} 
\label{pageref_subsubSection_Stirling_Polynomial_Forms} 

Several forms of the first triangle polynomials derived from the 
sequence of Stirling convolution polynomials are given in the 
following equations [cf. \sref{subsubSection_GenBernoulliPoly_ident_forms}]. 
%% : "FaSPoly_idents-2009.06.28.*": 
\begin{align} 
\label{eqn_Szt_form_for_FaPolys1} 
\FaSigmaPoly{\alpha}{n}{x} & = \sum_{k=0}^n 
     \frac{\alpha^k (1-\alpha)^{n-k}}{(n-k)!}\ \StirlingSigmaPoly{k}{x} \\ 
\label{eqn_FaSPoly_xp1_V1} 
\FaSigmaPoly{\alpha}{x-k}{x+1} & = \frac{k!}{x!}\ 
     \sum_{j=0}^{x-1} \gkpSI{x}{j+k} \binom{j+k}{k}\ \alpha^{x-k-j} 
     +\frac{1}{x!}\ \Iverson{k = 0} \Iverson{x \geq 0} \\ 
\label{eqn_FaSPoly_xp1_V2} 
   & = \frac{1}{x\phantom{!}}\ \sum_{j=0}^{x-1} \frac{(j+k)\ 
     \alpha^{x-k-j}}{j!}\ \StirlingSigmaPoly{x-k-j}{x} + \frac{1}{x!}\ 
     \Iverson{k = 0} \Iverson{x \geq 0}
\end{align} 
The generating series for the cases of diagonal indices in the 
original Stirling polynomial sequence is defined implicitly by the 
form of equation \eqref{eqn_Szt_form_for_FaPolys2} for a general real-valued 
parameter of $t$ \cite[\Section 6.2]{GKP} \cite{CVLPOLYS}. 
\begin{equation} 
\label{eqn_Szt_form_for_FaPolys2} 
\mathcal{S}_t(z) = \left(\frac{-\Log(1-z\mathcal{S}_t(z)^{t-1})}{z} 
     \right)^{1 / t} \implies [z^n]\mathcal{S}_t(z)^x = 
     x \StirlingSigmaPoly{n}{x+tn} 
\end{equation} 
Properties of the $\alpha$-factorial polynomial sequences in terms of the 
Stirling polynomials and the related generating series diagonals from 
equation \eqref{eqn_Szt_form_for_FaPolys2} are given by the 
next pair of identities [cf.\ \eqref{eqn_CM_7.50_GF_generalized}]. 
\begin{align} 
\label{eqn_Szt_form_for_FaPolys3} 
\FaSigmaPoly{\alpha}{n}{x+tn} & = [z^n]\left(\mathcal{S}_t(\alpha z)^x\ 
     \sum_{k=0}^n \frac{z^k \mathcal{S}_t(\alpha z)^k 
     (1-\alpha)^k}{(x+k)\ k!}\right) \\ 
\label{eqn_Szt_for_Fa2Polys1} 
\FaIISigmaPoly{\alpha}{n}{x+tn} & = \frac{(1-x+(1-t)n)}{(x+tn)} \sum_{k=0}^n 
     \frac{\alpha^k (x+(t-1)n)^k}{k!}\ \FaSigmaPoly{\alpha}{n-k}{1-x+(1-t)n} 
\end{align} 
The first identity in \eqref{eqn_Szt_form_for_FaPolys3} follows from the 
application of \eqref{eqn_Szt_form_for_FaPolys2} to the sum in equation 
\eqref{eqn_Szt_form_for_FaPolys1}. 
The second identity in \eqref{eqn_Szt_for_Fa2Polys1} follows from the 
variant polynomial form of $f_n^{\alpha}(x+(t-1)n)$ defined by equation 
\eqref{eqn_Fa2_fPoly_def_and_enum}. 
The identity in \eqref{eqn_Szt_for_Fa2Polys1} can also be expressed in 
terms of the implicit generating function of equation 
\eqref{eqn_Szt_form_for_FaPolys2} 
for the parameter value $t := -1$ through expansion of the sum in 
\eqref{eqn_Szt_form_for_FaPolys3}. 

As remarked in \sref{subSection_JFactPolys_Motivations_and_Background}, the 
Stirling polynomials are a special case of the finite-degree 
polynomial sequence forms defined and termed by Knuth as 
\quotetext{\keyword{convolution polynomials}}. The general 
convolution polynomial sequence form is defined by the 
series expansion of the generating function $F(z)^{x}$ corresponding to 
some (formal) power series $F(z)$ such that $F(0) = 1$ and where 
$x$ denotes the domain of polynomial variable \cite{CVLPOLYS}. 
The $\FaSigmaPoly{\alpha}{n}{x}$ polynomials differ from the 
required convolution polynomial generating function series form by 
an exponential multiplier in \eqref{Fa_MF_poly_GF_enum_exp}, though the 
$\alpha$-factorial polynomials satisfy similar 
convolution properties as given in the references \cite[cf.\ Table 272]{GKP}. 
In particular, the $\alpha$-factorial polynomial sequences can be 
expressed in terms of the identities given by following equations when 
the integer $\lvert \alpha \rvert > 1$. 
\begin{align*} 
\FaSigmaPoly{\alpha}{n}{x+y} & = \sum\limits_{i+j+k= n} 
     \frac{(\alpha-1)^i}{i!} \FaSigmaPoly{\alpha}{j}{x}
     \FaSigmaPoly{\alpha}{k}{y} \\ 
\FaSigmaPoly{\alpha}{n}{x_1+\cdots+x_k} & = 
     \sum\limits_{i_0+i_1+\cdots+i_k= n} 
     \frac{(k-1)^{i_0}(\alpha-1)^{i_0}}{i_0!} \FaSigmaPoly{\alpha}{i_1}{x_1} 
     \cdots \FaSigmaPoly{\alpha}{i_k}{x_k} \\ 
x n\ \FaSigmaPoly{\alpha}{n}{x+y} & = 
     \sum\limits_{i+j+k= n} \left(\frac{x\ y\ (\alpha-1)^i\ j}{i!}\ 
     \FaSigmaPoly{\alpha}{j}{x} \FaSigmaPoly{\alpha}{k}{y}\right) + 
     (\alpha-1)y\ \FaSigmaPoly{\alpha}{n-1}{x+y} 
\end{align*} 
These related identities can all be 
formally verified easily by performing simple operations on formal 
power series to the polynomial sequence generating generating functions. 
The special case when $\alpha := 1$ corresponding to the 
Stirling convolution polynomial sequence satisfies the 
original set of properties provided explicitly in the references 
\cite[Table 272]{GKP} \cite{CVLPOLYS}. 

\subsubsection{Additional Properties of the Polynomials of the First Kind} 

The property given by equation \eqref{eqn_Fa_pGF_Deriv_closedform_ident} for 
the first coefficient triangle in \eqref{Fa_rdef} results in the next pair of 
polynomial identities. 
%% : "FaSPoly_idents-2009.07.19-current.*": 
\begin{align*} 
\FaSigmaPoly{\alpha}{n}{x} & = [z^x] \left(\frac{\alpha^{n+1-x} z 
     (1+\alpha z)^{\frac{1}{\alpha}+x-1}}{\Log(1+\alpha z)^{n+1-x}} 
     + \frac{(x-1)\alpha z\ E_{n+1-x}\left((1-x-\alpha^{-1}) 
     \Log(1+\alpha z)\right)}{\alpha^{x-n}\ \Log(1+\alpha z)^{n-x}}\right) \\ 
   & \phantom{}= [z^x] \left( \frac{z \alpha^{n+1-x} 
     (1+\alpha z)^{\frac{1}{\alpha}+x-1}}{\Log(1+\alpha z)^{n+1-x}} 
     +\frac{(x-1) \alpha z\ \Gamma\left(x-n;\ (1-x-\alpha^{-1}) 
     \Log(1+\alpha z)\right)}{(\alpha-\alpha x-1)^{x-n}}\right) 
\end{align*} 
Next, the form of the enumerative identity in 
\eqref{eqn_CM_7.49_GF_generalized} is used to derive the 
polynomial form of the next equation in 
\eqref{eqn_FaSPoly_rev_x_input_identity}. 
\begin{equation} 
\label{eqn_FaSPoly_rev_x_input_identity} 
\FaSigmaPoly{\alpha}{n}{1-x} = \sum_{k=0}^{n+2x-2} 
     \frac{(x-1)!\ (-\alpha x)^{n+2x-2-k}}{(k+1-x)!\ (n+2x-2-k)!}\ 
     \FIIcfII{\alpha}{k+2-x}{x} 
\end{equation} 
Finally, the definition in equation \eqref{Fa2_MF_poly_defs1} 
combined with the form in equation \eqref{eqn_Fa2_in_terms_of_S2_sum} is 
used to derive the form of equation \eqref{eqn_Fa2_coeff_def_fnm}. 
Several polynomial-related identities for the second coefficient triangle 
\eqref{Fa2_rdef} are also defined in terms of the 
generating function coefficients of equation \eqref{eqn_Fa2_coeff_def_fnm} 
as given by the following equations. 
\begin{align} 
\label{eqn_Fa2_coeff_def_fnm} 
\FIIcfII{\alpha}{n}{m} & = \frac{(n-1)!}{(m-1)!}\ 
     [z^n] \left(z^m e^z \left(\frac{-\alpha z e^{-\alpha z}}{ 
     e^{-\alpha z}-1}\right)^{1-m}\right) \\ 
\notag 
   & = \frac{(n-1)!}{(m-1)!}\ 
     \sum_{j=0}^{n-m} \frac{(-\alpha)^j\ \StirlingSigmaPoly{j}{1-m}}{ 
     (n-m-j)!} \\ 
\notag 
   & = \frac{(n-1)!}{(m-1)!}\ \sum_{i+j+k= n-m} 
     \frac{(1-\alpha)^i\ (-\alpha)^{j+k}}{i!\ j!}\ B_j\ 
     \StirlingSigmaPoly{k}{-m} \\ 
\notag 
   & = \frac{(n-1)!}{(m-1)!}\ \sum_{j=0}^{n-m} \frac{(-1)^j\ 
     (2-\alpha)^{n-m-j}}{(n-m-j)!}\ \FaSigmaPoly{\alpha}{j}{1-m} \\ 
\notag 
   & = \frac{(n-1)!}{(m-1)!}\ \sum_{i+j+k = n-m} 
     \frac{2^i\ (-\alpha)^{i+j}}{i!\ j!}\ B_j\ \FaSigmaPoly{(-\alpha)}{k}{-m} 
\end{align} 
These properties are also related to the polynomials of the second kind from 
equation \eqref{Fa2_MF_poly_defs1} through the coefficient identities 
for the second triangle in \eqref{Fa2_rdef}. 

\section{Applications of the $j$-Factorial Polynomials} 
\label{Section_JFact_Poly_Applications} 

\subsection{$j$-Factorial Function Expansions} 
\label{subSection_MF_expansion_polys} 

The primary motivation in considering the 
generalized $\alpha$-factorial polynomial sequences is to 
formulate a rigorous method for expanding and enumerating forms of the 
$\alpha$-factorial functions, $(s-1)!_{(\alpha)}$, as 
polynomials of finite-degree in the formal input $s$. 
The $\alpha$-factorial function expansions discussed in this 
section are directly related to the author's continued work on 
the divisibility of both multifactorial and binomial coefficient expansion 
polynomials that are partitioned by indices of an integer-valued 
arithmetic progression of the form $\langle qn+r \rangle$, or 
equivalently of the class of $n \in \mathbb{N}$ such that 
$n \equiv r \pmod{q}$ for some non-trivial $q, r \in \mathbb{N}$. 

As an example, the initial motivation in considering the expansion 
coefficients of equation \eqref{Fa_rdef} involved finding generating 
function forms for the polynomial expansions of the form 
$\sum_{k} [s^{2k}] (s-1)!_{(2)}$ evaluated as the functions in $s$ 
corresponding to the distinct polynomial expansions of the double factorial 
function. More generally, the properties of the expansions of the forms 
$\sum_{k} [s^{qk+r}] (s-1)!_{(\alpha)}$ and 
$\sum_{k \in \mathbb{S}} [s^{k}] (s-1)!_{(\alpha)}$ for an 
arbitrary indexing set $\mathbb{S}$ motivate the next explorations of the 
generalized $\alpha$-factorial polynomial and coefficient triangle properties. 

Let the formal expansion polynomial variable for the 
$\alpha$-factorial functions be fixed as the notation $s$ and let the 
$\alpha$-factorial function expansion polynomials 
corresponding to the arbitrary indexing set 
$\mathbb{S} \subseteq \mathbb{N}$ be defined as follows: 
\begin{equation} 
\label{eqn_MFact_Exp_Polys_def} 
p_n^{\alpha}(s) := \sum_{k \in \mathbb{S}} 
     \FcfII{\alpha}{n}{k+1} (-1)^{n+k} s^{k}. 
\end{equation} 
The special case where $\mathbb{S} := \mathbb{N}$ corresponds the full 
$\alpha$-factorial function expansion of $(s-1)!_{(\alpha)}$. 
The next several sections generalize the forms of the expansions in 
\eqref{eqn_MFact_Exp_Polys_def} and establish other interesting properties 
related to these cases of the generalized $\alpha$-factorial 
function expansions. 

\subsubsection{The Primary Result for Binomial Coefficients} 

Given the expansion variable offset 
$s_0 \in [0,\ \alpha-1] \subseteq \mathbb{N}$ and the indexing set 
$\mathbb{S} \subseteq \mathbb{N}$, define the next generating functions for 
$\widehat{S}$ and $\widetilde{S}$ by the following pair of equations. 
\begin{align} 
\label{prop_MF_exps_GF_S} 
\widehat{S}(\alpha;\ x;\ z) & := \frac{z}{x}\ \widetilde{S}(z) e^{(1-\alpha) z} 
     {\left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)}^x \\ 
\label{eqn_Stilde_MFact_Exp_Polys_def} 
\widetilde{S}(z) & := \sum_{n=0}^{\infty} \frac{(-1)^n (s-s_0+1)^n 
     \Iverson{n \in \mathbb{S}}}{n!}\ z^n 
\end{align} 
The results of equations \eqref{eqn_prop_form1} and 
\eqref{eqn_prop_form2} define the 
$\alpha$-factorial polynomial expansions for the 
binomial coefficients \citeseqnumII{A007318}{A000984} 
\cite[(5.1); \Section 5]{GKP} involving the 
coefficient extractions corresponding to an 
arbitrary indexing set $\mathbb{S}$ with closed-form 
exponential generating function defined by 
equation \eqref{eqn_Stilde_MFact_Exp_Polys_def}. 
For the case where $\alpha \mid s-s_0$, the next forms give alternate 
symbolic expressions that can be used to rephrase the 
classical result given by the binomial theorem. 
\begin{align} 
\label{eqn_prop_form1} 
\binom{\frac{s-s_0}{\alpha}}{x} & = \frac{(x+1)}{\alpha^x} \sum_{k=0}^x 
     \frac{(-1)^k\ \FaSigmaPoly{\alpha}{k}{x+1}}{(x-k)!}\ (s-s_0)^{x-k} \\ 
\label{eqn_prop_form2} 
\sum_{n \in \mathbb{S}} [(s-s_0)^n] \binom{\frac{s-s_0}{\alpha}}{x} & = 
   \frac{(-1)^x (x+1)}{\alpha^x}\ [z^{x+1}] \widehat{S}(\alpha;\ x+1;\ z) 
\end{align} 
The form of equation \eqref{eqn_prop_form1} allows for the 
binomial coefficient expansions to be expressed in terms of the 
various formulations of the $\alpha$-factorial polynomial 
identities in \sref{Section_MF_Fa_polynomials}. 
In particular the identities of equations \eqref{eqn_FaSPoly_xp1_V1} and 
\eqref{eqn_FaSPoly_xp1_V2} may be applied to the binomial coefficient forms. 
A number of the enumerations from \sref{subSection_Fa_GF_form_idents} are 
also related to these expansions. 

Given the expansion identities and applications discussed in 
\sref{Section_Symm_poly_topics}, it is also interesting to consider that the 
Newton series expansion for the digamma function can be 
evaluated symbolically in terms of the binomial coefficients through the 
next identity. 
\begin{equation*} 
\PolyGamma{0}{s+1} = -\gamma_{\E}-\sum_{k=1}^{\infty} \binom{s}{k} 
     \frac{(-1)^{k}}{k}\ :=\ 
     \sum_{k=1}^{\infty} \sum_{j=0}^{k} c_{k, j} s^{j} 
\end{equation*} 
The coefficients $c_{k, j}$ defined implicitly by this expansion 
may of course then be considered in terms of the 
symbolic binomial forms in $s$ given by the results of this section. 

\subsubsection{Remarks on Special Cases of the 
               Indexing Sets $\mathbf{S \subseteq N}$} 
\label{subsubSection_Remarks_on_Indexing_Sets} 
\label{example_QF_tables} 

The full form of the generating function in 
equation \eqref{prop_MF_exps_GF_S} can be explicitly formulated for the 
set $\mathbb{S}$ by determining a specific closed-form for the 
generating function $\widetilde{S}(z)$ corresponding to the definition given 
for that particular indexing set. 
For example, let the formal expansion variable for the 
$\alpha$-factorial function be denoted by $s$ and consider the 
following useful special cases of the series indexing set 
$\mathbb{S} \subseteq \mathbb{N}$: 
\begin{align*} 
 & \mathbb{S} := {\langle 1 \rangle} \phantom{n+r} 
     \implies \widetilde{S}(z) = \sum_{n=0}^{\infty} 
     \frac{(-1)^{n} s^{n}}{n!}\ z^n = e^{-s z} \\ 
 & \mathbb{S} := {\langle qn+r \rangle} 
     \implies \widetilde{S}(z) = \sum_{n=0}^{\infty} 
     \frac{(-1)^{qn+r} s^{qn+r}}{(qn+r)!}\ z^n. 
\end{align*} 

More generally, suppose $\mathbb{G} = \langle g_n \rangle$ is a 
sequence defined over the natural numbers, $G(z)$ denotes the 
ordinary generating function for the set $\mathbb{G}$, 
$\widehat{G}(z)$ denotes the exponential generating function for $\mathbb{G}$, 
$\mathbb{A} := \langle qn+r \rangle$ is a prescribed 
arithmetic progression over the natural numbers, and 
$\omega_q := e^{2\pi\imath / q}$ denotes the 
\keyword{primitive $q^{th}$ root of unity}. 
The desired form of the generating function $\widetilde{S}(z)$ from equation 
\eqref{eqn_Stilde_MFact_Exp_Polys_def} can then expressed by the 
generating function $\widetilde{S}_{G}(z)$ through the following equations 
\cite[\Section 1.2.9]{TAOCP} \cite[\Section 2.4]{GFOLOGY}. 
%% : "OGF_EGF_forms_for_arithprog_indices.*": 
\begin{align*} 
\widetilde{S}_{G}(z) & := \phantom{\frac{1}{q}\ } \sum_{n=0}^{\infty} 
     \frac{(-s)^n\ g_n}{n!} \Iverson{n \in \mathbb{A}} z^n \\ 
   & \phantom{:} = \frac{1}{q}\ \sum_{k=0}^{q-1} \frac{1}{2 \pi} 
     \int_{-\pi}^{\pi} \omega_q^{-kr}\ G(-\omega_q^{k}\ e^{-\imath t} sz) 
     e^{e^{\imath t}} dt \\ 
   & \phantom{:} = \frac{1}{q}\ \sum_{k=0}^{q-1} 
     \omega_q^{-kr}\ \widehat{G}(-\omega_q^k\ sz) 
\end{align*} 
The second equation results from the fact that given the ordinary generating 
function for a sequence, the exponential generating function for that 
sequence may be obtained by the transformation identity 
\cite[p.\ 566;\ cf.\ p.\ 373]{GKP} 
\begin{equation} 
\label{eqn_OGF_to_EGF_transform} 
\widehat{G}(z) = \frac{1}{2\pi} \int_{-\pi}^{+\pi} G(z e^{-\imath t}) 
     e^{e^{\imath t}} dt, 
\end{equation} 
where the original generating function, $G$, is treated as an 
analytic function over its domain. 
The following examples serve to illustrate the utility in the 
form of these special case variations on the set $\mathbb{S}$. 

As an example, 
define the exponential generating function corresponding to the complete 
expansion forms of the \keyword{quadruple-factorial} function for 
$\alpha := 4$ in the formal variable $s$ by equation \eqref{Ex_5.1_eqn1}. 
\begin{equation} 
\label{Ex_5.1_eqn1} 
\widehat{S}(x;\ z) = \frac{z}{x}\ e^{-(3+s) z} 
     \left(\frac{4z e^{4 z}}{e^{4 z}-1}\right)^x 
\end{equation} 
The full expansion forms for $(s-1)!_{(4)}$ and the corresponding range of 
offset values corresponding each distinct polynomial expansion of the function 
are summarized in \tref{table_4Fact_full_exp_polys} where the 
form of the generating function $\widehat{S}(z)$ is defined by 
equation \eqref{Ex_5.1_eqn1} \citeseqnumIV{A034176}{A000407}{A007696}{A047053}. 
\bigskip 
\begin{numberedtable}[Full-Indexed Quadruple-Factorial Expansions] 
\label{table_4Fact_full_exp_polys} 
\begin{center} 
\begin{tabular}{|l|l|r|r|r|r|} 
\hline 
$x$ & $x!\ (-1)^{x+1}\ [z^x] \widehat{S}(x;\ z)$ & $s \mapsto 4x$ & 
      $s \mapsto 4x-1$ & $s \mapsto 4x-2$ & $s \mapsto 4x-3$ \\  
\hline 
1 & 1 & 1 & 1 & 1 & 1 \\
2 & $(s-1)$ & 7 & 6 & 5 & 4 \\
3 & $(s-1)(s-5)$ & 77 & 60 & 45 & 32 \\
4 & $(s-1)(s-5)(s-9)$ & 1155 & 840 & 585 & 384 \\
%5 & $s^4-28 s^3+254 s^2-812 s+585$ & 21945 & 15120 & 9945 & 6144 \\
%6 & $s^5-45 s^4+730 s^3-5130 s^2+14389 s-9945$ & 
%    504735 & 332640 & 208845 & 122880 \\ 
\hline 
\end{tabular} 
\end{center} 
\end{numberedtable} 
\vskip 0.2in 
Next, consider the expansions of the function $(s-1)!_{(4)}$ 
corresponding to the distinct even-indexed and 
odd-indexed coefficient sets enumerated by the 
respective generating functions in equations \eqref{Ex_5.1_eqn2} and 
\eqref{Ex_5.1_eqn3}. 
\begin{align} 
%% : "check_4Fact_EvenOddGF_ex_coeff_idents.*": 
\label{Ex_5.1_eqn2} 
\widehat{S}_{2n}(x;\ z) & := 
      \sum_{k \geq 0}\left(\sum_{j \geq 0} \FcfII{4}{x}{2j+1} 
      \frac{(-s)^{2j}}{x!}\right) z^k \phantom{^{+1}} = \phantom{-} 
      \frac{z}{x}\ \cosh(s z)\ e^{-3z} 
      \left(\frac{4z e^{4 z}}{e^{4 z}-1}\right)^x \\ 
\label{Ex_5.1_eqn3} 
\widehat{S}_{2n+1}(x;\ z) & := 
      \sum_{k \geq 0}\left(\sum_{j \geq 0} \FcfII{4}{x}{2j+2} 
      \frac{(-s)^{2j+1}}{x!}\right) z^k = -\frac{z}{x}\ \sinh(s z)\ 
     e^{-3 z} \left(\frac{4z e^{4 z}}{e^{4 z}-1}\right)^x 
\end{align} 
These particular forms of the generating function from equation 
\eqref{eqn_Stilde_MFact_Exp_Polys_def} follow from the cases of the 
series variable transformations that result from the function 
$\frac{1}{2}\left(e^{-sz} \pm e^{sz}\right)$ \cite[cf.\ \Section 7.2]{GKP}. 
A comparison of the even-indexed, odd-indexed, and complete polynomial 
expansions of the quadruple-factorial function is provided in 
\tref{table_4Fact_EvenOdd_exp_polys}. 
\bigskip 
\begin{numberedtable}[Alternately-Indexed Quadruple-Factorial Expansions] 
\label{table_4Fact_EvenOdd_exp_polys} 
\begin{center} 
\begin{tabular}{|l|l|l|l|} 
\hline 
$x$ & $x!\ (-1)^{x+1} [z^x] \widehat{S}(x;\ z)$ & 
      $x!\ (-1)^{x+1} [z^x] \widehat{S}_{2n}(x;\ z)$ & 
      $x!\ (-1)^{x+1} [z^x] \widehat{S}_{2n+1}(x;\ z)$ \\ 
\hline 
%1 & 1 & 1 & 0 \\
2 & $s-1$ & $-1$ & $s$ \\
3 & $s^2-6 s+5$ & $s^2+5$ & $-6 s$ \\
4 & $s^3-15 s^2+59 s-45$ & $-15 s^2-45$ & $s^3+59 s$ \\
5 & $s^4-28 s^3+254 s^2-812 s+585$ & $s^4+254 s^2+585$ & $-28 s^3-812 s$ \\
%6 & s^5-45 s^4+730 s^3-5130 s^2+14389 s-9945 & -45 s^4-5130 s^2-9945 & s^5+730 s^3+14389 s
\hline 
\end{tabular} 
\end{center} 
\end{numberedtable} 
\bigskip 

\subsubsection{Arithmetic Progressions of Expansion Coefficients} 

For the indexing set $\mathbb{S} := \Nset$, the 
$\alpha$-factorial expansion polynomials 
are described succinctly by the recurrence relation in equation 
\eqref{eqn_pn_SeqN_rform} for the 
polynomials $\bar{p}_n^{\alpha}(s)$ and by the resulting closed-form 
recurrence solution in equation \eqref{eqn_pn_SeqN_GammaFn_form}. 
\begin{align} 
\label{eqn_pn_SeqN_rform} 
\bar{p}_n^{\alpha}(s) & = (s-1+(1-n)\ \alpha)\ \bar{p}_{n-1}^{\alpha}(s) 
     +\Iverson{n = 0} \\ 
\label{eqn_pn_SeqN_GammaFn_form} 
   & = (-\alpha)^{n}\ \Gamma\left(n+\frac{1-s}{\alpha}\right) 
       \Gamma\left(\frac{1-s}{\alpha}\right)^{-1} 
\end{align} 
In general, the expansion polynomials corresponding to the indexing set 
$\mathbb{S}$ are of an arbitrary finite integral degree where the form 
of the resulting expansion sequences cannot be determined recursively 
in such simple terms as that of \eqref{eqn_pn_SeqN_rform}. 
That being said, for the particular form of the indexing set 
defined by the arithmetic progression $\mathbb{A} = \langle qn+r\rangle$, 
a similar technique to that of \sref{subsubSection_Remarks_on_Indexing_Sets}
can be applied in formulating the expansion polynomials 
corresponding to the generalized analog of equation 
\eqref{eqn_pn_SeqN_GammaFn_form}. The polynomial expansions corresponding to 
this choice of $\mathbb{S} := \mathbb{A}$ can be expressed through the form of 
equation \eqref{eqn_pn_phat_rootsofunity_sum_form} 
\cite[\Section 1.2.9]{TAOCP}. 
\begin{equation} 
\label{eqn_pn_phat_rootsofunity_sum_form} 
p_n^{\alpha}(s) = \frac{1}{q}\ \sum_{k=0}^{q-1} 
     e^{-2\pi\imath kr/q}\ \bar{p}_n^{\alpha}\left(e^{2\pi\imath k/q}\ s\right) 
\end{equation} 

As an example, consider the indices corresponding to the 
arithmetic progression defined by the set $\mathbb{S} := \langle 3n \rangle$ 
consisting of all non-negative integer multiples of $3$ \citeseqnum{A008585}. 
Let $\omega$ denote the \keyword{primitive cube root of unity}. 
The following identities give the exact form of the polynomials defined by 
equation \eqref{eqn_pn_phat_rootsofunity_sum_form} for this 
specific progression as well as the exponential generating function 
corresponding to the expansions over these indices. 
\begin{align*} 
%p_n^{\alpha}(s) \phantom{\frac{z^n}{n!}} & = 
%     \frac{(-1)^n a^n}{3} \left(\frac{\Gamma 
%     \left(\frac{a n-s+1}{a}\right)}{\Gamma\left((1-s)/a\right)}+ 
%     \frac{\Gamma \left(\frac{a n-(-1)^{2/3} s+1}{a}\right)}{\Gamma
%     \left((1-(-1)^{2/3} s)/a\right)}+\frac{\Gamma 
%     \left(n+\frac{\sqrt[3]{-1} s+1}{a}\right)}{\Gamma\left((\sqrt[3]{-1}\ 
%     s+1)/a\right)}\right) \\ 
%\sum_{n=0}^{\infty} p_n^{\alpha}(s) \frac{z^n}{n!} & = 
%     \frac{(a z+1)^{-\frac{\sqrt[3]{-1} s+1}{a}}}{3} 
%     \left((a z+1)^{\frac{\imath \sqrt{3} s}{a}}+(az+1)^{\frac{\left( 
%     1+\sqrt[3]{-1}\right) s}{a}}+1\right) 
% 
%% : "Section6.1.3_omega_substs.*": 
p_n^{\alpha}(s) \phantom{\frac{z^n}{n!}} & = 
     \frac{(-1)^n \alpha^n}{3} \left(\frac{\Gamma 
     \left(n+\frac{1-s}{\alpha}\right)}{\Gamma\left(\frac{1-s}{\alpha} 
     \right)}+\frac{\Gamma \left(n+\frac{1-\omega s}{\alpha}\right)}{\Gamma
     \left(\frac{1-\omega s}{\alpha}\right)}+\frac{\Gamma 
     \left(n+\frac{1-\omega^2 s}{\alpha}\right)}{\Gamma\left( 
     \frac{1-\omega^2 s}{\alpha}\right)}\right) \\ 
\sum_{n=0}^{\infty} p_n^{\alpha}(s) \frac{z^n}{n!} & = 
     \frac{(1+\alpha z)^{-\frac{1-\omega^2 s}{\alpha}}}{3} 
     \left((1+\alpha z)^{\frac{\imath \sqrt{3} s}{\alpha}}+(1+\alpha z)^{ 
     \frac{\left(1-\omega^2\right) s}{\alpha}}+1\right) 
\end{align*} 

While the form of equation \eqref{eqn_pn_phat_rootsofunity_sum_form} works well 
for computing the expansion polynomial forms for specific given cases of an 
indexing arithmetic progression, a more useful and 
general result can be derived by considering the exponential generating 
function of the form in equation \eqref{eqn_pn_SeqN_GammaFn_form}. 
%% : "MFact_exp_polys_idents-2008.07.31-poly_EGFs_and_OGF_forms.*": 
Define the exponential generating function and 
analogous property corresponding to the 
expansions of equations \eqref{eqn_pn_SeqN_rform} and 
\eqref{eqn_pn_SeqN_GammaFn_form} as follows: 
\begin{align*} 
\widehat{p}(z) & := \sum_{n=0}^{\infty} \frac{\bar{p}_n^{\alpha}(s)}{n!}\ 
     z^n = (1+\alpha z)^{\frac{(s-1)}{\alpha}} 
     \phantom{;}= \sum_{k=0}^{\infty} \frac{\Log(1+\alpha z)^{k}}{ 
     (1+\alpha z)^{1/\alpha}\ \alpha^{k}\ k!}\ s^k \\ 
S_{\mathbb{A}}(s;\ z) & := \sum_{k \in \mathbb{A}} [s^k] \widehat{p}(z)\ 
     k!\ s^k = \frac{\alpha^{q-r}\ s^r\ \Log(1+\alpha z)^{r}}{ 
     (1+\alpha z)^{1 / \alpha}\ \left(\alpha^q-s^q\ \Log(1+\alpha z)^{q} 
     \right)}. 
\end{align*} 
In order to compute the expansion polynomial forms defined by 
equation \eqref{eqn_MFact_Exp_Polys_def} for the arbitrary progression 
$\mathbb{S} = \mathbb{A}$, it is necessary to obtain the 
exponential generating series for the 
function $S_{\mathbb{A}}(s;\ z)$ in terms of the formal parameter $s$. 
It follows from the last equations and the 
transformation result of \eqref{eqn_OGF_to_EGF_transform} that the 
identity in the next equation holds for the 
original sequence of expansion polynomials, $p_n^{\alpha}(s)$. 
\begin{align*} 
\frac{p_n^{\alpha}(s)}{n!} & = \sum_{k=0}^{n} [s^k z^n] 
     S_{\mathbb{A}}(s;\ z)\ \frac{s^k}{k!} = [z^n] \left( 
     \frac{1}{2\pi}\ \int_{-\pi}^{\pi} S_{\mathbb{A}}(s e^{-\imath t};\ z)\ 
     e^{e^{\imath t}}\ dt\right) 
\end{align*} 

Additional enumerations for the indexing set 
$\mathbb{S} := \langle qn+r \rangle$ are based on the 
results of equations \eqref{eqn_pn_SeqN_GammaFn_form} and 
\eqref{eqn_pn_phat_rootsofunity_sum_form}. 
The derivation of these results provides additional 
insight in to the expansion polynomial forms of \eqref{eqn_MFact_Exp_Polys_def} 
assumed over the arithmetic progression of indices where in these cases the 
expansion variable $s$ is treated as a \emph{non-formal} series index. 

%% : "MFact_exp_polys_idents-2008.07.31-poly_EGF_ts_forms.*": 
First, observe that 
\eqref{eqn_pn_phat_rootsofunity_sum_form} results from the form of 
\eqref{eqn_pn_SeqN_GammaFn_form} as the defined sum of polynomial terms of the 
general form $e_{k}\ \bar{p}_{n}^{\alpha}(w_k s)$ \cite{TAOCP}. 
It follows from this observation that 
\begin{equation*} 
\sum_{n=0}^{\infty} e_{k}\ \bar{p}_n^{\alpha}(w_k s)\ \frac{z^n}{n!} 
     = e_{k}\ (1+\alpha z)^{(sb+c) / \alpha} 
\end{equation*} 
where $b := w_k$, $c := -1$, and the coefficients $e_{k}$ are 
complex-valued scalars. 
For the $\alpha$-factorial function, the polynomial expansion parameter $s$ 
is defined numerically by a specific subset of values over the 
natural numbers corresponding to each $\alpha$ 
[cf.\ \sref{subSection_Fa_Fa2_prelim_defs};\ 
\tref{table_4Fact_full_exp_polys}]. 
Assume that the input is defined as the arbitrary index $s \in \Nset$ and 
form the generating series over the index through the following equation. 
\begin{equation*} 
\bar{p}(t;\ z) := \sum_{s=0}^{\infty} (1+\alpha z)^{(sb+c) / \alpha} t^s = 
     \frac{(1+\alpha z)^{b / \alpha}}{1-t (1+\alpha z)^{c / \alpha}} 
\end{equation*} 
Next, let the sequence of polynomials, $\widehat{p}_k$, given in terms of the 
implicit variable $t$ be defined as in the next equation. 
\begin{equation*} 
\widehat{p}_k := \frac{p_k}{k!} = [\alpha^{0} z^k] \bar{p}(t;\ z) 
     = [z^k] \left(\frac{e^{b z}}{1-e^{c z} t}\right) 
\end{equation*} 
The identities for the coefficients in the series expansion of the function 
$\bar{p}(t;\ z)$ can be expressed from the form of the last equation as 
follows: 
\begin{align*} 
[z^n] \bar{p}(t;\ z) & = \sum_{k=1}^{n} \gkpSI{n}{k}\ \frac{k!}{n!}\ 
     (-1)^{n+k}\ \alpha^{n-k}\ \widehat{p}_k 
   = \sum_{k=1}^{n} (-1)^{n+k} \alpha^{n-k}\ 
     \StirlingSigmaPoly{n-k}{n} k\ \widehat{p}_k \\ 
   & = \sum_{k=0}^{\infty} \left( 
     \sum_{j=0}^{n} \frac{(b+ck)^{j}\ \alpha^{n-j}\ B_{n-j}^{(n)}}{n\ 
     (j-1)!\ (n-j)!}\ \Iverson{n \geq 1}+\Iverson{n=0}\right) t^k. 
\end{align*} 
The given coefficient properties then allow the 
expansion polynomial terms in equation 
\eqref{eqn_pn_phat_rootsofunity_sum_form} to be enumerated by a series in 
$t$ over the polynomial domain of $s \in \Nset$ and in terms of the 
discrete sequence index $n$. The resulting series form in 
equation \eqref{eqn_pbar_coeff_in_terms_of_HurwitzLerchPhi_fn} 
follows from the last coefficient identity 
where $\Phi(z;\ s;\ a) := \sum_{n=0}^{\infty} (n+a)^{-s}\ z^n$ denotes the 
\keyword{Lerch transcendent} function. 
\begin{equation} 
\label{eqn_pbar_coeff_in_terms_of_HurwitzLerchPhi_fn} 
[z^n] \bar{p}(t;\ z) = 
     \sum_{j=0}^{n} \frac{c^j \alpha^{n-j}\ B_{n-j}^{(n)}}{n\ 
     (j-1)!\ (n-j)!}\ \Phi\left(t;\ -j;\ 
     {\frac{b}{c}}\right)\ \Iverson{n \geq 1}+\frac{\Iverson{n = 0}}{1-t} 
\end{equation} 
It is then possible to 
extend the enumeration result defined over the polynomial index $s \in \Nset$ 
to the polynomial expansions originally defined by equation 
\eqref{eqn_MFact_Exp_Polys_def} for the particular indexing arithmetic 
progression as follows: 
\begin{equation} 
\label{eqn_pns_exp_polys_enum_GFnt_over_s_var} 
\sum_{s=0}^{\infty} p_n^{\alpha}(s) t^s = 
     \sum_{k=0}^{q-1} \sum_{j=0}^{n} 
     \frac{(-1)^{j}\ \omega_q^{-kr}\ \alpha^{n-j}\ B_{n-j}^{(n)}}{ 
     q\ n\ (j-1)!\ (n-j)!}\ \Phi(t;\ -j;\ -\omega_q^{k}) \Iverson{n \geq 1} + 
     \frac{\Iverson{n = 0}}{1-t}. 
\end{equation} 
The generating series of equation 
\eqref{eqn_pns_exp_polys_enum_GFnt_over_s_var}
results from the initial conditions of the recurrence in 
\eqref{eqn_pn_SeqN_rform} and by combining the forms of 
\eqref{eqn_pn_phat_rootsofunity_sum_form} and 
\eqref{eqn_pbar_coeff_in_terms_of_HurwitzLerchPhi_fn} where 
$\omega_q := e^{2\pi\imath / q}$ again denotes the 
primitive $q^{th}$ root of unity. 

%% : "MFact_exp_polys_idents-2008.07.06-Pz_GF_forms.*": 
Consider the generating function in 
\eqref{eqn_CM_7.50_GF_generalized_V1} and the next series expansion 
result given by equation \eqref{eqn_formal_series_exp_in_s}. 
\begin{equation} 
\label{eqn_formal_series_exp_in_s} 
\sum_{k=1}^{\infty} \frac{(k-1+z) \alpha^{\Nc} z^{\Nc+k-1} 
     e^{(\alpha \Nc+1)z} }{(k-1)! \left(e^{\alpha z}-1\right)^{\Nc}}\ 
     s^{k-1} = (s+1) e^{s z}\left(\frac{\alpha^{\Nc} z^{\Nc+1}\ 
     e^{(\alpha \Nc+1)z}}{\left(e^{\alpha z}-1\right)^{\Nc}}\right) 
\end{equation} 
Let the shorthand notation $\omega := e^{2\pi\imath / q}$ denote the 
primitive $q^{th}$ root of unity. 
For the indexing set $\mathbb{S} := \mathbb{A}$, the 
following interesting expansion form is based on the result of 
\eqref{eqn_formal_series_exp_in_s} for $q, n \in \Nset$ and 
where $q \geq 2$ and $n \geq 1$ \cite[\Section 1.2.9]{TAOCP}. 
\begin{equation*} 
%[z^n]\ \sum _{k=0}^{q-1} \frac{\left(1+s e^{\frac{2 i \pi  k}{q}} 
%     \right)}{q} e^{s z e^{\frac{2 i \pi  k}{q}}-\frac{2 i\pi  k r}{q}} 
%     = \frac{s^n \left(e^{2 i \pi r}-1\right) 
%     \left(s e^{\frac{2 i \pi n}{q}}+ 
%     e^{\frac{2 i \pi  (n+1)}{q}}-s e^{\frac{2 i \pi  r}{q}}- 
%     e^{\frac{2 i \pi  r}{q}}\right)}{n!\ q\ e^{\frac{2 i \pi  (q-1) r}{q}} 
%     \left(e^{\frac{2 i \pi  n}{q}}-e^{\frac{2 i \pi  r}{q}}\right) 
%     \left(e^{\frac{2 i \pi r}{q}}-e^{\frac{2 i \pi  (n+1)}{q}}\right)} 
[z^n]\left(\sum_{k=0}^{q-1} \left(1+s \omega^k\right) \omega^{-kr} 
     e^{sz \omega^k}\right) = 
     \frac{\left(\omega^{qr}-1\right) \left((s+\omega)\ \omega^n-(s+1)\ 
     \omega^r\right)}{\omega^{(q-1)r} \left(\omega^n-\omega^r\right) 
     \left(\omega^r-\omega^{n+1}\right)} \cdot \frac{s^n}{n!} 
\end{equation*} 
The exponential generating function defined next in equation 
\eqref{eqn_Pz_GF_def} results from the expansion identity of the 
last equation. 
\begin{equation} 
\label{eqn_Pz_GF_def} 
\widehat{P}_{n}(z) := \frac{\alpha^n z^{n+1} e^{(\alpha n+1) z}}{q\  
     \left(e^{\alpha z}-1\right)^{n}}\ \sum_{n=0}^{\infty} 
     \frac{\left(\omega^{qr}-1\right) \left((s+\omega)\ \omega^n-(s+1)\ 
     \omega^r\right)}{\omega^{(q-1)r} \left(\omega^n-\omega^r\right) 
     \left(\omega^r-\omega^{n+1}\right)} \cdot \frac{(sz)^n}{n!} 
\end{equation} 
Finally, for positive integer $n$ and the given indexing progression 
$\mathbb{S}$, the expansion polynomials of equation 
\eqref{eqn_MFact_Exp_Polys_def} are then expressed in terms 
of equation \eqref{eqn_Pz_GF_def} by the series coefficients 
\begin{equation*} 
\frac{p_{n+1}^{\alpha}(s)}{n!} = [z^n] \widehat{P}_{n}(z). 
\end{equation*} 

\subsubsection{An Exponential Generating Series for $\mathbf{S \subseteq N}$} 

For the more general indexing set 
$\mathbb{S} \subseteq \mathbb{N}$, define the polynomial form 
$b_n(s)$ and the corresponding exponential generating functions for the 
sequence by equation \eqref{eqn_bj_and_BHat_EGF_deriv_defs}. 
\begin{equation} 
\label{eqn_bj_and_BHat_EGF_deriv_defs} 
b_j(s) := \sum_{k \in \mathbb{S}} \binom{j}{k} s^k \implies %\mapsto 
     \widehat{B}(z) := \sum_{j=0}^{\infty} \frac{b_{j}(s)}{j!}\ z^j 
     \implies \widehat{B}^{\prime}(z) = 
     \sum_{j=0}^{\infty} \frac{b_{j+1}(s)}{j!}\ z^j 
\end{equation} 
The identity of equation \eqref{eqn_Fa_in_terms_of_S1_sum_V1} yields the 
next pair of identities for the general expansion polynomial form in 
\eqref{eqn_MFact_Exp_Polys_def} as follows: 
\begin{align} 
\notag 
p_n^{\alpha}(s) & = \sum_{j=0}^{n-2} \gkpSI{n-1}{j+1} 
     (-1)^{n+1} \alpha^{n-2-j}\ b_{j+1}(-s) \\ 
\label{eqn_pn_exp_poly_Bz_GF_seriescoeffs} 
\frac{(-1)^{n+1}}{(n-2)!}\ p_n^{\alpha}(-s) & = 
     [z^n]\left(z^2 \widehat{B}^{\prime}(z) 
     \left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)^{n-1}\right). 
\end{align} 
The generating functions enumerating the sequence of the $b_j(s)$ polynomials 
have a special form for the case when the indexing set is 
defined as the arithmetic progression $\mathbb{S} := \langle qn+r \rangle$. 
The generating function for this special case is given as equation 
\eqref{eqn_arithprog_set_Bz_GF}. 
\begin{align} 
\label{eqn_arithprog_set_Bz_GF} 
B(z) & := \sum_{j=0}^{\infty} \sum_{k=0}^j \binom{j}{qk+r} s^{qk+r} 
       z^n = \frac{(-s z)^r (z-1)^{q-1-r}}{-(1-z)^{q}+(sz)^q} 
%\label{eqn_arithprog_set_BHatz_EGF} 
%\widehat{B}(z) & \phantom{:}= \frac{1}{2\pi}\ 
%     \int_{-\pi}^{\pi} B(z e^{-\imath t})\ e^{e^{\imath t}} dt 
\end{align} 
The exponential generating function corresponding to the special case and in 
equation \eqref{eqn_pn_exp_poly_Bz_GF_seriescoeffs} then follows from the 
transformation given in \eqref{eqn_OGF_to_EGF_transform}. 
Additional identities for \eqref{eqn_pn_exp_poly_Bz_GF_seriescoeffs} 
result from expanding the right hand side generating series in 
terms of the generalized Bernoulli polynomials, Stirling polynomials, 
alternate $\alpha$-factorial expansion forms, and the related results of 
\sref{Section_MF_Fa_polynomials}. 

\subsubsection{Recurrence Relations and Ordinary Differential Equations for 
               $\mathbf{S \subseteq N}$} 

Let $\mathbb{S} \subseteq \Nset$ denote an 
arbitrary indexing set for the polynomial expansions considered by 
equation \eqref{eqn_MFact_Exp_Polys_def} and let the closed-form 
exponential generating function $\widetilde{S}$ be defined by the 
next equation. 
\begin{equation*} 
\widetilde{S}(z) := \sum_{n \in \mathbb{S}} \frac{(-1)^{n}\ s^{n}}{n!}\ z^n 
\end{equation*} 
The $\alpha$-factorial expansion polynomials for general the indexing set 
$\mathbb{S}$ can be defined through the 
following identities where the form of the 
series coefficients in equation \eqref{eqn_pn_GF_SeriesCoeff_def_form_V2} 
follows from equations \eqref{eqn_prop_form1} and 
\eqref{Fa_poly_internal_idents1} and from the generating function in 
equation \eqref{Fa_MF_poly_GF_enum_exp}. 
\begin{align} 
\notag 
\frac{(-1)^{n}}{n!} p_{n+1}^{\alpha}(s) & = 
     [z^n] \widetilde{S}(z) S_{\alpha}(n+1;\ z) \\ 
\label{eqn_pn_GF_SeriesCoeff_def_form_V2} 
p_{n+1}^{\alpha}(s) & = \sum_{k=0}^{x} [z^k] \widetilde{S}(z) 
     \frac{1}{(x+1)^2} \left(k\ \FaSigmaPoly{\alpha}{x-k}{x}+ 
     (\alpha x+1-\alpha)\ \FaSigmaPoly{\alpha}{x-1-k}{x}\right)
\end{align} 
For $x \in \Nset$, the expansion polynomials satisfy the recurrence 
relation in \eqref{eqn_px_rform_RHS} where the 
generating function $R(x;\ z)$ is defined by the first equation 
\eqref{eqn_px_rform_RxzFn}. 
\begin{align} 
\label{eqn_px_rform_RxzFn} 
\phantom{[z^x]} R(x;\ z) & = {\alpha^{x-1} z^{x+1} 
     e^{(1+(x-1) \alpha) z} \left(e^{\alpha z}-1\right)^{1-x}} 
     \left(\widetilde{S}^{\prime\prime}(z)+\widetilde{S}^{\prime}(z)\right) \\ 
\label{eqn_px_rform_RHS} 
[z^x] R(x;\ z) & = (x-1)x(x+1)\ p_{x+1}^{\alpha}(s)- 
     (x-1)x(\alpha x+1-\alpha)\ p_{x}^{\alpha}(s) 
\end{align} 
Let the next pair of ordinary generating functions be defined as 
follows: 
\begin{equation*} 
R(z) := \sum_{x=0}^{\infty} [z^x] R(x;\ z)\ z^x 
     \quad \text{ and } \quad 
P(z) := \sum_{x=0}^{\infty} p_x^{\alpha}(s)\ z^x. 
\end{equation*} 
It follows from the given recurrence relations for the 
expansion polynomials in \eqref{eqn_MFact_Exp_Polys_def} that the 
generating function defined by the form of $P(z)$ satisfies the 
ordinary differential of equation \eqref{eqn_px_GF_Pz_diffEQ}. 
\begin{equation} 
\label{eqn_px_GF_Pz_diffEQ} 
P^{(3)}(z)+\left(\frac{\alpha+1}{\alpha z-1}+\frac{3}{z}\right) 
     P^{\prime\prime}(z)+ 
     \frac{2(\alpha+1)}{z(\alpha z-1)} P^{\prime}(z) 
     +\frac{p_0^{\alpha}(s)}{z^3 (\alpha z-1)} 
     = \frac{R(z)}{z^3 (1-\alpha z)} 
\end{equation} 

\subsubsection{Application to Wilson's Theorem and Related Identities} 

The classical result of \keyword{Wilson's theorem} can be 
rephrased in terms of the $\alpha$-factorial expansions considered by 
this section. Let the positive integer $p$ assume the form $p := ak+b$ for 
$a, b \in \mathbb{N}$. 
Wilson's theorem states the result of the next equation \cite[(4.49)]{GKP}. 
\begin{equation} 
\label{eqn_WilsonsTheorem} 
(ak+b-1)! = -1 \pmod{p} \quad \iff \quad \text{$p$ is prime} 
\end{equation} 
The following derivation gives the symbolic expansion of the factorial 
function from the original theorem of equation 
\eqref{eqn_WilsonsTheorem} in terms of the 
coefficients of the $\alpha$-factorial triangle in \eqref{Fa_rdef} and the 
previous results of the section. 
\begin{align*} 
(as+b-1)! & = \prod_{m=0}^{n-1} (as-am+b-1) = 
     \sum_{j=0}^{n} \FcfII{a}{n+1}{j+1} (-1)^{n+j}\ (as+b)^{j} \\ 
   & = (-1)^{n}\ \Gamma(n+2)\ [z^{n+1}]\left( 
     \frac{z}{(n+1)} e^{-(as+b)z} e^{(1-a) z} \left(\frac{a z e^{a z}}{ 
     e^{a z}-1}\right)^{n+1}\right) 
\end{align*} 
It follows that for $n := \lceil(as+b-1)/a\rceil \longmapsto s$, the 
form of the input factorial function to Wilson's theorem may be 
expanded as in equation \eqref{eqn_WThm_LHS_FactFns_GF_coeff_def}. 
\begin{equation} 
\label{eqn_WThm_LHS_FactFns_GF_coeff_def} 
(as+b-1)! = (-1)^{s}\ \Gamma(s+1)\ [z^s]\left(e^{-(as+b)z} 
     e^{(1-a) z} \left(\frac{a z e^{a z}}{e^{a z}-1}\right)^{s+1}\right) 
\end{equation} 

Additional results related to Wilson's theorem can be handled using related 
techniques for the $\alpha$-factorial function expansions. 
Consider the result of \keyword{Clement's theorem} as stated by equation 
\eqref{eqn_ClementsThm_mods} concerning the modular properties of 
\keyword{twin prime} clusters \cite{CLEMENTTPRIMES}. 
\begin{equation} 
\label{eqn_ClementsThm_mods} 
4\left((p-1)!+1\right) + p \equiv 0 \pmod{p(p+2)} \quad \iff \quad 
     p,\ p+2 \text{ prime } 
\end{equation} 
Let the function $a: \mathbb{N} \rightarrow \mathbb{Z}$ be 
defined such that $a(p)$ assumes a positive integral value if and 
only if $p$ and $p+2$ are both prime and is zero-valued otherwise. 
It follows from this definition and from equation \eqref{prop_MF_exps_GF_S} 
for $\widetilde{S}(z) := e^{-p}$ that the next generating function 
identities for the theorem statement of \eqref{eqn_ClementsThm_mods} hold 
if and only if the variable index $p$ corresponds to a 
twin prime pair of the form $(p,\ p+2)$. 
\begin{align*} 
%% : "*-Clements_in_multifact.*": 
\frac{a(p)(p^2+2p)}{p!} & = [z^pt^p]\left(\frac{(z+4)t e^{z}}{1-t} - 
     4z\Log\left(1-\frac{e^{z}zt}{e^{z}-1}\right)\right) \\ 
\frac{a(p)(p^2+2p)}{p!} & = [z^pt^p]\left(\frac{(p+4)t e^{z}}{1-t} + 
     \frac{4z^2t e^{pz}}{p(e^{z}-1-zt)}\right) 
\end{align*} 

These new expansions allow for the modular residues required for 
primality by the theorems in equations \eqref{eqn_WilsonsTheorem} and 
\eqref{eqn_ClementsThm_mods} to be considered analytically in terms of the 
enumerations of \sref{Section_MF_Fa_polynomials}, including the 
full arsenal of modular properties documented in the referenced literature 
for the generalized (higher-order) Bernoulli polynomials. 
An abundance of properties giving the generalized Bernoulli polynomials, 
N\"{o}rlund polynomials, and classical Stirling number triangles in terms of 
equivalent forms modulo primes, prime powers, and 
other bases are derived in the 
referenced works of Adelberg and a lengthy list of other authors 
\cite{UNIVERSALBPCGS,ADELBERGBPFACTORS,ADELBERGp-ADICBNUMCGS,
       NORLUNDARITHPROPS,CONGRUENCESNNUMSANDB2NUMS,NORLUNDPOLYPROPS, 
       STIRLINGPOLYS,FTHOWEXPLDEGEN,CGSBPEP,MWARDFACTSUMS,
       PTYOUNGDEGEN,PTYOUNGCGSBESNUMS} 
\cite[\Section 6, \Section 8, and \Section 9]{ADELBERGA} 
\cite[\Section 3.2]{HS}. Additionally, the determination of the 
largest power of an arbitrary prime dividing the 
factorial function and the applications of these bounds is discussed in 
the references by Adelberg and the works of several other authors 
\cite[cf.\ \Section 2, (3), (4), and (5)]{ADELBERGp-ADICBNUMCGS} 
\cite{LEGENDRESIDENT} \cite[\Section 4.4]{GKP}. 
Results concerning the residues and properties of the 
expansions identified for the theorems of this section are another topic 
considered by the author's continued research and exploration of these forms. 

\subsection{An Extension of Stirling's Approximation and 
            Asymptotics Related to the Generalized $j$-Factorial Functions} 
\label{subSection_StirlingsApprox_generalized} 

\subsubsection{Preliminaries} 

The primary values of the $\alpha$-factorial functions considered by this 
section are defined by the function in equation 
\eqref{eqn_MFax_values_fn_def}. 
\begin{align} 
\label{eqn_MFax_values_fn_def} 
\MF_{\alpha}(x) & := \prod_{m=0}^{x-1} (1+m\ \alpha) = 
     x!\ [z^x] (1-\alpha z)^{-1/\alpha} = \FcfII{\alpha}{x+1}{1}
\end{align} 
A table of the $\MF_{\alpha}(x)$ function values corresponding to the 
first several $\alpha$ parameters is provided in 
\tref{table_MF_function_value_summary} 
\citeseqnumVI{A00142}{A001147}{A007559}{A007696}{A008548}{A008542}. 
\bigskip 
\begin{numberedtable}[The $\mathbf{MF_{a}(x)}$ Function Values] 
\label{table_MF_function_value_summary} 
\begin{center} 
\begin{tabular}{|c|r|r|r|r|r|r|r|} 
\hline 
$x$ & $\MF_1(x)$ & $\MF_2(x)$ & $\MF_3(x)$ & $\MF_4(x)$ & $\MF_5(x)$ & 
      $\MF_6(x)$ & $\MF_7(x)$ \\ \hline 
$1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
$2$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
$3$ & 6 & 15 & 28 & 45 & 66 & 91 & 120 \\
$4$ & 24 & 105 & 280 & 585 & 1056 & 1729 & 2640 \\
$5$ & 120 & 945 & 3640 & 9945 & 22176 & 43225 & 76560 \\
$6$ & 720 & 10395 & 58240 & 208845 & 576576 & 1339975 & 2756160 \\
$7$ & 5040 & 135135 & 1106560 & 5221125 & 17873856 & 49579075 & 118514880 \\
$8$ & 40320 & 2027025 & 24344320 & 151412625 & 643458816 & 2131900225 & 5925744000 \\ 
\hline 
\end{tabular} 
\end{center} 
\end{numberedtable} 
 
\bigskip 
The asymptotic formula of \keyword{Stirling's approximation} for the 
single factorial function is a well-known classical result that is 
restated for reference and clarity as equation 
\eqref{eqn_StirlingsApprox_SA_form}. 
\begin{equation} 
\label{eqn_StirlingsApprox_SA_form} 
x! \sim \sqrt{2 \pi x}\ \left(\frac{x}{e}\right)^{x} := \SA(x) 
\end{equation} 
Stirling's original result is discussed in his 
book \booktitle{Methodus Differentialis}\nocite{STIRLINGMD}. 
Works that are related to the methods employed in generalizing 
Stirling's formula include several items in the references 
by B\"{u}hring and others \cite{PRODGAMMAFNRATIO,GAMMAFNRATIO} 
\cite[\Section 9.6]{GKP}. 
A historical summary of approaches to and proofs concerning 
Stirling's formula and the closely-related \keyword{Stirling series} is 
also detailed in the work of Davis \cite{GAMMAFNHISTORY} and 
Dominici \cite{STIRLINGTHEMESVARS}. 

The $\alpha$-factorial function approximated by the 
asymptotic result derived in the next section is defined by the function in 
equation \eqref{eqn_MFax_values_fn_def}. 
While this function offers the most intuitive 
set of values for the integer-valued $\alpha$-factorial functions, the 
expansions corresponding to each individual $\alpha$-factorial 
triangle in \eqref{Fa_rdef} may differ by an offset, or shift, in the 
expansion variable [cf.\ \sref{subSection_Fa_Fa2_prelim_defs};\ 
\tref{table_4Fact_full_exp_polys}]. 
Specifically, for $s_0 \in [-1,\ \alpha+2]$, the 
alternate function values may be computed from a specific offset as the 
product $\prod_{m=1}^{x-1} (1+s_0+m\ \alpha)$. 
The ratio of the shifted $\alpha$-factorial function to the 
prescribed factorial function form in \eqref{eqn_MFax_values_fn_def} is 
denoted as the function $R(x)$ given by the next equation. 
\begin{equation*} 
R(x) := \prod_{m=1}^{x-1} \left(1+\frac{s_0}{1+m\alpha}\right) 
      = \frac{\Gamma(1+\frac{1}{\alpha})\ \Gamma(x+\frac{s_\SmallZero+1}{ 
        \alpha})}{\Gamma(1+\frac{s_\SmallZero+1}{\alpha})\ 
        \Gamma(x+\frac{1}{\alpha})} 
\end{equation*} 
It follows from the given product that the values of 
$R(x)$ can be approximated by 
Stirling's approximation formula in \eqref{eqn_StirlingsApprox_SA_form} in the 
form of the next equation. 
\begin{equation} 
\label{eqn_Rx_SA_approx_form} 
R(x) \sim \frac{e^{-\frac{s_\SmallZero}{\alpha}} \left(\frac{1}{\alpha} + 
     x\right)^{-\frac{1}{\alpha}-x} \Gamma \left(1+\frac{1}{\alpha}\right) 
     \left(\frac{s_\SmallZero+1}{\alpha}+x\right)^{\frac{s_\SmallZero}{\alpha} 
     +\frac{1}{\alpha}+x} \sqrt{\frac{\alpha x+s_\SmallZero+1}{\alpha}}}{ 
     \sqrt{\frac{\alpha x+1}{a}} \Gamma \left(1+\frac{s_\SmallZero+1}{\alpha} 
     \right)}
\end{equation} 
While the consideration is unnecessary in the 
original form for the single factorial function where $\alpha := 1$, in the 
general case for increasing $\alpha \geq 2$, the ratio of the functions 
$(s-1)!_{(\alpha)} / (s+s_0-1)!_{(\alpha)}$ may in fact differ significantly 
from unity for the approximation in the limit as $s \rightarrow \infty$. 
The approximation for $R(x)$ considered by equation 
\eqref{eqn_Rx_SA_approx_form} 
allows for the shifted $\alpha$-factorial function variants to be 
approximated by the results derived next in 
\sref{subsubSection_SAGen_ResultDeriv} for the 
function offset defined by equation \eqref{eqn_MFax_values_fn_def}. 

\subsubsection{Derivation of the Generalized Approximation Formula} 
\label{subsubSection_SAGen_ResultDeriv} 

Suppose that $\langle h_{\alpha} \rangle$ is a sequence of positive real 
numbers, $a_1 \neq b_1$, and $c$ is a fixed positive real scalar. 
For sufficiently large $\alpha$, equation \eqref{GFLECT_Lemma_4.8_restated} 
restates the content of Lemma $4.8$ in Lando's book concerning the 
asymptotic approximations of certain hypergeometric sequences 
\cite[\Section 4.2]{GFLECT}. 
\begin{equation} 
\label{GFLECT_Lemma_4.8_restated} 
\frac{h_{\alpha+1}}{h_{\alpha}} = A\ 
     \frac{\alpha^{k}+a_1\alpha^{k-1}+\cdots+a_{k}}{
     \alpha^{k}+b_1\alpha^{k-1}+\cdots+b_{k}} \implies 
     h_{\alpha} \sim c\ A^{\alpha}\ \alpha^{a_1-b_1} 
\end{equation} 
The lemma is the inspiration for generalizing 
Stirling's approximation for the 
single factorial function to approximate the rates of growth of the 
$\alpha$-factorial function cases defined by equation 
\eqref{eqn_MFax_values_fn_def} for the case of integer $\alpha \geq 2$. 
 
The forms of \sref{subSection_MF_expansion_polys} can be 
reformulated as a sequence over the integer index $\alpha$ as 
parametrized by the formal expansion variable $s$ (also $x$). From the 
results of equations \eqref{eqn_prop_form1} and \eqref{eqn_prop_form2} the 
required coefficient form in the lemma is 
\begin{equation*} 
(a_1-b_1)\ s^{x-1} = x\ (\FaSigmaPoly{\alpha+1}{1}{x+1}- 
     \FaSigmaPoly{\alpha}{1}{x+1})\ s^{x-1} = 
     \frac{1}{2}\ x(x-1)\ s^{x-1}. 
\end{equation*} 
It follows from equation \eqref{GFLECT_Lemma_4.8_restated} that the 
form of the $x^{th}$ distinct expansion of the $\alpha$-factorial function, 
$(s-1)!_{(\alpha)}$, in the formal input $s$ is approximated by the form 
\begin{equation*} 
a_s(x) \sim c_x s^{x(x-1)/2} 
\end{equation*} 
where the constant $c_x$ is strictly a function of the expansion index $x$ 
and is entirely independent of the fixed $s$. 
To study the form assumed by the constant, consider the following forms: 
\begin{align*} 
\bar{c}_x & := (s-1-\alpha(x-2))\ \bar{c}_{x-1}+\Iverson{x = 1} \\ 
\widehat{c}_{x} & := \frac{\bar{c}_{x}}{\sqrt{s}^{(x^2-x)}} = 
     \frac{(-\alpha)^{x-1}\ s^{(x-x^2)/2}\ \Gamma\left(x-1 + 
     \frac{1-s}{\alpha}\right)}{\Gamma\left(\frac{1-s}{\alpha}\right)} \\ 
   & \phantom{:}= \sum_{k=1}^{x-1}\sum_{j=0}^{k} \gkpSI{x-1}{k}\ 
     \binom{k}{j}\ (-1)^{x-1+j}\ \alpha^{x-1-k}\ s^{(x-x^2)/2+j}. 
\end{align*} 
For $r := s^{x}$ and $b := \alpha\ (s/r)^{1/2}$, 
the expression for the coefficient results in the next equation of 
\eqref{eqn_chatx_BPoly_form}. 
\begin{align} 
\notag 
\sum_{x=0}^{\infty} \widehat{c}_x\ t^x = & 
     \frac{(s-1)}{\alpha^2}\ b^2 t^2\ e^{\frac{(s-1)}{\alpha} bt}\ 
     \left(\frac{bt}{e^{bt}-1}\right)^{x-1} \\ 
\label{eqn_chatx_BPoly_form} 
  & \implies \widehat{c}_x = \alpha^{x-2}\ (s-1)\ s^{(x-x^2)/2}\ 
     B_{x-2}^{(x-1)}\left(\frac{s-1}{\alpha}\right) 
\end{align} 
For each distinct expansion indexed by $x$, the 
formal expansion parameter $s$ corresponding to the form of equation 
\eqref{eqn_MFax_values_fn_def} is given by the 
particular value $s := (\alpha x+2-\alpha)$. The generalized Bernoulli 
polynomial identity in \eqref{eqn_chatx_BPoly_form} has the 
special case expansion form given by equation 
\eqref{eqn_SAGenApprox_BP_SpecCaseExpansion} 
\cite[cf.\ (16)]{ADELBERGp-ADICBNUMCGS} 
\cite[\keyword{N\"{o}rlund Numbers}]{CONGRUENCESNNUMSANDB2NUMS} 
\cite{BPOLYSOLDANDNEW}. 
\begin{align} 
\label{eqn_SAGenApprox_BP_SpecCaseExpansion} 
B_{x-2}^{(x-1)}\left(x-1+\frac{1}{\alpha}\right) 
   & = \frac{\Gamma\left(x-1+\frac{1}{\alpha}\right)}{
     \Gamma\left(1+\frac{1}{\alpha}\right)} \approx \frac{\SA\left(x-2+
     \frac{1}{\alpha}\right)}{\Gamma\left(1+\frac{1}{\alpha}\right)} 
\end{align} 
As discussed by Graham et.\ al.\ \cite[\Section 9.6]{GKP}, 
Stirling's approximation also holds for the non-integer inputs that 
result from the Bernoulli polynomial expansion of the last equation. 
The approximate form of $\widehat{c}_x$ then follows from the 
Bernoulli polynomial approximation in 
\eqref{eqn_SAGenApprox_BP_SpecCaseExpansion} and from the form of equation 
\eqref{eqn_chatx_BPoly_form}. 

For $a_s(x) \mapsto \MF_{\alpha}(x)$, 
Stirling's approximation formula is generalized to the 
more general $\alpha$-factorial function where $\alpha \geq 2$ as the 
result expressed in equation \eqref{eqn_GenStirlingsApprox_MTilde_form}. 
The denominator factor of $\Gamma(1/\alpha)$ in the approximation is 
a constant dependent on the prescribed parameter $\alpha$ for the 
functions approximated by the result. 
\begin{equation} 
\label{eqn_GenStirlingsApprox_MTilde_form} 
\widetilde{M}_{x}^{\alpha} = 
     \frac{\sqrt{2 \pi }\ \alpha^{x+\frac{1}{2}} e^{-\frac{1}{\alpha}-x+2}  
     \left(x-2+\frac{1}{\alpha}\right)^{\frac{1}{\alpha}+x} 
     (\alpha x-\alpha+1)}{(\alpha x-2 \alpha+1)^{3/2}\ \Gamma 
     \left(\frac{1}{\alpha}\right)} 
\end{equation} 
The entire form of the $\alpha$-factorial function cases defined by 
\eqref{eqn_MFax_values_fn_def} for the positive integer $\alpha \geq 1$ is 
approximated by the identity of equation \eqref{eqn_GenStirlingsApprox}. 
\begin{equation} 
\label{eqn_GenStirlingsApprox} 
\MF_{\alpha}(x) \sim \SA(x) \Iverson{\alpha = 1} 
     + \widetilde{M}_{x}^{\alpha} \Iverson{\alpha \geq 2}. 
\end{equation} 

\subsubsection{Relative Error in the Generalized Approximations} 

Let the relative error in the approximations of equations 
\eqref{eqn_StirlingsApprox_SA_form} and 
\eqref{eqn_GenStirlingsApprox_MTilde_form} be 
defined in respective order by the functions $\varepsilon_{\SA}$ and 
$\varepsilon_{\MF}$ such that 
\begin{equation*} 
\varepsilon_{\SA}(x) := \left\lvert\frac{SA(x)}{x!}- 
     \frac{x!}{\SA(x)}\right\rvert \quad \text{ and } \quad 
\varepsilon_{\MF}(\alpha;\ x) := \left\lvert\frac{
     \widetilde{M}_{x}^{\alpha}}{\MF_{\alpha}(x)}-\frac{\MF_{\alpha}(x)}{ 
     \widetilde{M}_{x}^{\alpha}}\right\rvert. 
\end{equation*} 
The relative error functions computed for each of the approximation formulas 
results appear to be comparable and are summarized for the 
first several inputs of $x$ in \tref{table_SA_approx_rel_error_summary}. 
It stands to reason that if the empirical observation holds, the 
error in computing the $\alpha$-factorial functions from the 
approximate result in \eqref{eqn_GenStirlingsApprox_MTilde_form} may be 
estimated from some known variant of the Stirling series coefficients, such as 
in the forms of established error terms considered classically in the 
asymptotic series for the single factorial function. 
\bigskip 
\begin{numberedtable}[Relative Error in the Approximation Results] 
\label{table_SA_approx_rel_error_summary} 
\begin{center} 
\begin{tabular}{|c|r|r|r|r|r|r|r|} 
\hline 
$x$ & $\varepsilon_{\SA}(x)$ & $\varepsilon_{\MF}(2;\ x)$ & 
      $\varepsilon_{\MF}(3;\ x)$ & $\varepsilon_{\MF}(4;\ x)$ & 
      $\varepsilon_{\MF}(5;\ x)$ & $\varepsilon_{\MF}(6;\ x)$ \\ \hline 
$1$ & 0.1623010 & 1.9027600 & 1.7507200 & 1.6964300 & 1.7025100 & 1.7427600 \\
$2$ & 0.0827049 & 0.3080580 & 0.4368270 & 0.5517920 & 0.6558050 & 0.7510340 \\
$3$ & 0.0553629 & 0.1096830 & 0.1230140 & 0.1309570 & 0.1362300 & 0.1399840 \\
$4$ & 0.0415843 & 0.0663379 & 0.0710266 & 0.0736272 & 0.0752806 & 0.0764245 \\
$5$ & 0.0332909 & 0.0474968 & 0.0498588 & 0.0511299 & 0.0519240 & 0.0524673 \\
$6$ & 0.0277531 & 0.0369790 & 0.0383966 & 0.0391469 & 0.0396113 & 0.0399271 \\
$7$ & 0.0237940 & 0.0302711 & 0.0312150 & 0.0317094 & 0.0320135 & 0.0322196 \\
$8$ & 0.0208229 & 0.0256216 & 0.0262948 & 0.0266449 & 0.0268594 & 0.0270043 \\
$9$ & 0.0185112 & 0.0222096 & 0.0227137 & 0.0229745 & 0.0231339 & 0.0232413 \\
$10$ & 0.0166613 & 0.0195991 & 0.0199908 & 0.0201925 & 0.0203155 & 0.0203984 \\ 
\hline 
\end{tabular} 
\end{center} 
\end{numberedtable} 
\bigskip 
The application of \keyword{Euler's summation formula} as 
it applies to the original form of Stirling's series for 
$\Log(n!)$ is an important topic of interest considered in 
\booktitle{Concrete Mathematics} \cite[\Section 9.5 and \Section 9.6]{GKP}. 
The discussion of Stirling's approximation for $n!$ in the 
reference leads to a similar derivation of the more general identity for the 
approximation of equation \eqref{eqn_MFax_values_fn_def}. 
The result given by equation 
\eqref{eqn_LogMFax_EulerSummationForm_gen_in_CM_9.6} 
follows from the instructive treatment and the remarks provided in the 
reference \cite[\Section 9.6: pp.\ 479--481]{GKP}
where $\varphi_{m, n} \in (0,\ 1) \subseteq \mathbb{R}$, $B_k$ is an 
element of the classical sequence of Bernoulli numbers 
\citeseqnumII{A000367}{A002445}, and 
$\alpha \in [2,\ \infty) \subseteq \Nset$ 
\cite[\Section 3]{GENBNUMSEULERSUMS} \cite[\Section 5]{PROPSr-STIRSERIES}. 
\begin{align} 
\label{eqn_LogMFax_EulerSummationForm_gen_in_CM_9.6} 
\Log(\MF_{\alpha}(x)) & = \sum_{1 \leq k < x} \Log(1+\alpha k) = 
     (1+x) \Log(1+x)+(1-x)-\Log(4)+\sigma_{C} \\ 
\notag & \phantom{=} 
     +\sum_{k=1}^{m} \frac{\alpha^{2k-1}\ B_{2k}}{2k(2k-1) 
     (1+\alpha x)^{2k-1}}+\frac{\alpha^{2m+1}\ 
     B_{2m+2}\ \varphi_{m, n}}{(2m+2)(2m+1)(1+\alpha x)^{2m+1}} 
\end{align} 
The constant $\sigma_{C}$ from the last equation may be treated as a 
function of $\alpha$ and serves as a generalized analog to the 
form of \keyword{Stirling's constant}, $\sigma = \frac{1}{2} \ln(2\pi)$, 
derived explicitly for the single factorial case in the 
reference \cite[\Section 9.6]{GKP}. 
The form of \eqref{eqn_LogMFax_EulerSummationForm_gen_in_CM_9.6} is 
intended to suffice as an incomplete start to rigorously determining the 
unknown error term bounds for the approximation in equation 
\eqref{eqn_GenStirlingsApprox_MTilde_form}. 

\subsubsection{Example: Eulerian Triangle Row Sums} 
\label{subsubSection_ApplTo_EulerianTriangleRowSums} 

The row sums of the Eulerian number triangles provide an 
immediate application of the combined generalized approximation 
result in equation \eqref{eqn_GenStirlingsApprox} for an 
arbitrary positive integer-valued $\alpha$. The 
row sums of the first and \quotetext{second-order} Eulerian number 
triangles \citeseqnumII{A066094}{A008517} can be 
expressed in terms of products of the $\alpha$-factorial function 
expansions for the respective cases where $\alpha = 1$ and $\alpha = 2$ 
\cite[cf.\ (6.42) and \Section 6.2]{GKP}. 
The given properties of the well-known Eulerian number triangles suggest the 
generalization defined by the recurrence relation of the next equation 
\cite[cf.\ (6.35) and (6.41)]{GKP}. 
\begin{equation} 
\label{eqn_mthOrderEulerianTriangle_rdef} 
\gkpEj{n}{k}{m} := (k+1)\gkpEj{n-1}{k}{m}+(mn-k-m+1)\gkpEj{n-1}{k-1}{m} 
     + \Iverson{n = k = 0} 
\end{equation} 
The row sums formed by the generalized 
\keyword{$m^{th}$-order Eulerian triangles} from the last equation 
may then be expressed through the following identity: 
\begin{equation*} 
%\sum_{k=0}^{n-1} \gkpEj{n}{k}{m} = (mn-m+1)(mn-m) \cdots 1 
%     = (mn-m+1)!_{(m)} 
\sum_{k=0}^{n-1} \gkpEj{n}{k}{m} = \prod_{j=1}^{n-1} (1+m\ j) 
     = \MF_{m}(n). 
\end{equation*} 
For sufficiently large row index $n$, the corresponding row sums of the 
triangle in \eqref{eqn_mthOrderEulerianTriangle_rdef} may then be 
approximated by the full form of equation \eqref{eqn_GenStirlingsApprox} for 
each parameter $m \in [1,\ \infty) \subseteq \Nset$. 

\section{Acknowledgments} 
\label{Section_thanks_and_ack} 

The author thanks the anonymous referee of an earlier 
draft of the article for many useful bibliography references 
on related work and also Professor Bruce Reznick for 
suggestions in preparing the final draft of the manuscript. 

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\bigskip
\hrule
\bigskip

\noindent 2010 \emph{Mathematics Subject Classification}: 
Primary 05A10; Secondary 11B73, 11B83. 

\noindent\emph{Keywords}: 
Multifactorial, factorial, j--factorial, double factorial, 
factorial polynomial, Stirling number, Stirling triangle, 
Stirling number of the first kind, Stirling polynomial, 
Pochhammer symbol, Concrete Mathematics, 
Bernoulli polynomial, Bernoulli number, N\"{o}rlund polynomial, 
triangular recurrence, triangle, harmonic number, harmonic series, 
symmetric polynomial, binomial theorem, binomial coefficient, Newton series, 
Stirling's approximation, Stirling's formula, Methodus Differentialis, 
arithmetic progression, root of unity, exponential generating function, 
double generating function, super generating function, 
gamma function, polygamma function, exponential integral, 
modified Bessel function, Lerch transcendent, 
Euler--Mascheroni constant, Euler gamma, 
Eulerian number, Eulerian triangle, Wilson's theorem, Laguerre polynomial, 
umbral calculus, polynomial sequence, Appell sequence, Euler polynomial, 
Hermite polynomial. 

\bigskip
\hrule
\bigskip 

\noindent (Concerned with sequences \seqnum{A000079},
\seqnum{A000108},
\seqnum{A000110},
\seqnum{A000142},
\seqnum{A000165},
\seqnum{A000254},
\seqnum{A000367},
\seqnum{A000392},
\seqnum{A000399},
\seqnum{A000407},
\seqnum{A000454},
\seqnum{A000984},
\seqnum{A001008},
\seqnum{A001147},
\seqnum{A001296},
\seqnum{A001297},
\seqnum{A001298},
\seqnum{A001620},
\seqnum{A001813},
\seqnum{A002445},
\seqnum{A002805},
\seqnum{A006882},
\seqnum{A007318},
\seqnum{A007406},
\seqnum{A007407},
\seqnum{A007408},
\seqnum{A007409},
\seqnum{A007559},
\seqnum{A007661},
\seqnum{A007662},
\seqnum{A007696},
\seqnum{A008275},
\seqnum{A008276},
\seqnum{A008277},
\seqnum{A008278},
\seqnum{A008292},
\seqnum{A008297},
\seqnum{A008517},
\seqnum{A008542},
\seqnum{A008543},
\seqnum{A008544},
\seqnum{A008545},
\seqnum{A008546},
\seqnum{A008548},
\seqnum{A008585},
\seqnum{A027641},
\seqnum{A027642},
\seqnum{A032031},
\seqnum{A034176},
\seqnum{A045754},
\seqnum{A045755},
\seqnum{A047053},
\seqnum{A048993},
\seqnum{A048994},
\seqnum{A052562},
\seqnum{A066094},
\seqnum{A080417},
\seqnum{A081051},
\seqnum{A094638},
\seqnum{A098777},
\seqnum{A111593},
\seqnum{A130534}, and
\seqnum{A154959}.)


\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received July 21 2009;
revised version received  June 19 2010.
Published in {\it Journal of Integer Sequences}, June 21 2010.

\bigskip
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\noindent
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