%\documentstyle[twoside, amsmath, amssymb, amsfonts, eucal, epsfig, 12pt]{article} \documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf A Symbolic Operator Approach to Power \\ \vskip .1in Series Transformation-Expansion Formulas } \vskip 1cm \large Tian-Xiao He\footnote{The research of this author was partially supported by Artistic and Scholarly Development (ASD) Grant and sabbatical leave of the IWU.} \\ Department of Mathematics and Computer Science \\ Illinois Wesleyan University\\ Bloomington, IL 61702-2900 \\ USA\\ \href{mailto:the@iwu.edu}{\tt the@iwu.edu} \\ \end{center} \vskip .2in \begin{abstract} In this paper we discuss a kind of symbolic operator method by making use of the defined Sheffer-type polynomial sequences and their generalizations, which can be used to construct many power series transformation and expansion formulas. The convergence of the expansions are also discussed. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{defin}[theorem]{Definition} \newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}} \newtheorem{examp}[theorem]{Example} \newenvironment{example}{\begin{examp}\normalfont\quad}{\end{examp}} \newtheorem{remar}[theorem]{Remark} \newenvironment{remark}{\begin{remar}\normalfont\quad}{\end{remar}} \newcommand\pa{\parallel} \newcommand\wh{\widehat} \newcommand\wt{\widetilde} \newcommand\lp{L^p(\mathbb R)} \newcommand\bs{\backslash} \newcommand\ov{\overline} \newcommand\la{\langle} \newcommand\ra{\rangle} \newcommand\ri{\rightharpoonup} \newcommand\be{\begin{equation}} \newcommand\ee{\end{equation}} \newcommand\bn{\begin{eqnarray}} \newcommand\en{\end{eqnarray}} \newcommand\bns{\begin{eqnarray*}} \newcommand\ens{\end{eqnarray*}} \newcommand\bC{{\mathbb C}} \newcommand\bR{{\mathbb R}} \newcommand\bN{{\mathbb N}} \newcommand\bQ{{\mathbb Q}} \newcommand\bZ{{\mathbb Z}} \newcommand\bT{{\mathbb T}} \newcommand\bF{{\mathbb F}} \newcommand\Cs{{{\mathbb C}^s}} \newcommand\Rs{{{\mathbb R}^s}} \newcommand\Ns{{{\mathbb N}^s}} \newcommand\Zs{{{\mathbb Z}^s}} \newcommand\Ts{{{\mathbb T}^s}} \newcommand\cA{{\cal A}} \newcommand\cB{{\cal B}} \newcommand\cD{{\cal D}} \newcommand\cR{{\cal R}} \newcommand\cN{{\cal N}} \newcommand\cO{{\cal O}} \newcommand\cQ{{\cal Q}} \newcommand\cS{{\cal S}} \newcommand\cZ{{\cal Z}} \newcommand\cH{{\cal H}} \newcommand\tu{{\bigtriangleup}} \newcommand\td{{\bigtriangledown}} \def\d{{\rm d}} \def\a{\alpha} \def\l{\lambda} \section{Introduction} The closed form representation of series has been studied extensively. See, for examples, Comtet \cite{Comtet74}, Ch. Jordan \cite{Jordan65}, Egorechev \cite{Egorychev84}, Roman-Rota \cite{RomanRota78}, Sofo \cite{Sofo03}, Wilf \cite{Wilf94}, Petkov\v{s}ek-Wilf-Zeilberger's book \cite{Petkovsek96}, and the author's recent work with Hsu, Shiue, and Toney \cite{HHST05}. This paper is a sequel to the work \cite{HHST05} and the paper with Hsu and Shiue \cite{HHS073}, in which the main results are special cases of Theorem~\ref{thm:2.1} shown below. The object of this paper is to make use of the following defined generalized Sheffer-type polynomial sequences and the classical operators $\Delta$ (difference), $E$ (shift), and $D$ (derivative) to construct a method for the summation of power series expansions that appears to have a certain wide scope of applications. As an important tool in the calculus of finite differences and in combinatorial analysis, the operators $E$, $\Delta$, $D$ are defined by the following relations. \[ Ef(t)=f(t+1), \quad \Delta f(t)=f(t+1)-f(t), \quad Df(t)=\frac{d}{dt}f(t). \] Powers of these operators are defined in the usual way. In particular for any real numbers $x$, one may define $E^xf(t)=f(t+x)$. Also, the number $1$ may be used as an identity operator, viz. $1f(t)\equiv f(t)$. Then it is easy to verify that these operators satisfy the formal relations ({\it cf}. \cite{Jordan65}) \[ E=1+\Delta =e^D,\quad \Delta=E-1=e^D-1,\quad D=\log (1+\Delta). \] Note that $E^k f(0)=\left[ E^k f(t)\right]_{t=0}=f(k)$, so that $(xE)^kf(0)=f(k)x^k$. This means that $(xE)^k$ with $x$ as a parameter may be used to generate a general term of the series $\sum^\infty_{k=0}f(k)x^k$. \begin{definition}\label{def:1.0} Let $A(t)$, $B(t)$, and $g(t)$ be any formal power series over the real number field ${\bR}$ or complex number field ${\bC}$ with $A(0)=1,$ $B(0)=1$, $g(0)=0$, and $g'(0)\not=0$. Then the polynomials $p_n(x)$ ($n=0,1,2,\cdots$) defined by the generating function ($GF$) \be\label{eq:1.1} A(t)B(xg(t))=\sum_{n\geq 0}p_n(x)t^n \ee are called generalized Sheffer-type polynomials associated with $(A(t), B(t),$ $g(t))$. Accordingly, $p_n(D)$ with $D\equiv d/dt$ is called Sheffer-type differential operator of degree $n$ associated with $A(t)$, $B(t)$, and $g(t)$. In particular, $p_0(D)\equiv I$ is the identity operator due to $p_0(x)=1$. \end{definition} In Definition \ref{def:1.0}, if $B(t)=e^t$, then the defined $\{p_n(x)\}$ is a classical Sheffer-type polynomial sequence associated with $(A(t), g(t))\equiv (A(t),$ $exp(t), g(t))$. As examples, classical Sheffer-type polynomials include Bernoulli polynomials, Euler polynomials, and Laguerre polynomials generalized by $(A(t), g(t))=(t/(e^t-1), t)$, $(2/(e^t+1), t)$, and $((1-t)^{-p}, t/(t-1))$ ($p>0$), respectively. We call the infinite matrix $\left[d_{n,k}\right]_{n,k\geq 0}$ with real entries or complex entries a generalized Riordan matrix (the originally defined Riordan matrices need $g'(0)=1$) if its $k$th column satisfies \be\label{eq:1.2} \sum_{n\geq 0}d_{n,k}t^n=A(t)(g(t))^k; \ee that is, \[ d_{n,k}=[t^n]A(t)(g(t))^k, \] the $n$th term of the expansion of $A(t)(g(t))^k$. The Riordan matrix is denoted by $(A(t), g(t))$ or $[d_{n,k}]$ described in (\ref{eq:1.2}). Then the generalized Sheffer-type polynomial sequence associated with $(A(t), B(t), g(t))$ is the result of the following matrix multiplication \[ [d_{n,k}]\left[ \begin{array}{c} 1\\b_1x\\b_2x^2\\\vdots \\ b_nx^n\\ \vdots \end{array}\right]. \] If $\left[d_{n,k}=[t^n]A(t)(g(t))^k\right]$ and $\left[c_{n,k}=[t^n]C(t)(f(t))^k\right]$ are two Riordan matrices, and $\{p_n(x)\}$ and $\{q_n(x)\}$ are two corresponding generalized Sheffer-type polynomial sequences associated with $(A(t), B(t), g(t))$ and $(C(t), B(t), f(t))$, respectively, then we can define a umbral composition ({\it cf}. its special case of $B(t)=e^t$ is given in \cite{Roman84} and \cite{RomanRota78}) between $\{ p_n(x)\}$ and $\{q_n(x)\}$, denoted by $\{p_n(x)\}\# \{q_n(x)\}$. The resulting sequence is the generalized Sheffer-type polynomial sequence associated with $(A(t)C(g(t)),$ $B(t), f(g(t))$. Clearly, the sequence $\{p_n(x)\}\# \{q_n(x)\}$ is the result of the following matrix multiplication \[ [d_{n,k}][c_{n,k}]\left[ \begin{array}{c} 1\\b_1x\\b_2x^2\\\vdots \\ b_nx^n\\ \vdots \end{array}\right]. \] A power series $B(t)=1+\sum^\infty_{k=1}b_kt^k$ is said to be {\it regular} if $b_k\not=0$ for all $k\geq 1$. Under the composition operator $\#$, it can be proved that all generalized Sheffer-type polynomial sequences associated with a regular $B(t)$ form a group, called the generalized Sheffer group associated with $B(t)$. Its verification is analogous to the classical Sheffer group associated with $e^t$ established in \cite{Roman84} (see also in \cite{HHS072}). In \cite{HHS072}, the author established the isomorphism between the classic Sheffer group and the Riordan group based on the following bijective mapping: $\theta: [d_{n,k}]\mapsto \{p_n(x)\}$ or $\theta: (A(t),g(t))\mapsto \{p_n(x)\}$, i.e., \be\label{eq:1.3} \theta ([d_{n,k}]_{n\geq k\geq 0}):=\sum^n_{j=0}d_{n,j}x^j/j!=[d_{n,k}]_{n\geq k\geq 0}X, \ee for fixed $n$, where $X=(1,x, x^2/2!, \ldots)^T$, or equivalently, \be\label{eq:1.4} \theta((A(t), g(t)):=[t^n]A(t)e^{xg(t)} \ee It is clear that $(1,t)$, the identity Riordan array, maps to the identity Sheffer-type polynomial sequence $\{ p_n(x)\equiv x^n/(n!)\}_{n\geq 0}$. From the definitions (\ref{eq:1.2}) we immediately know that \be\label{eq:1.5} p_n(x)=[t^n]A(t)e^{x g(t)}\,\,\, \text{if and only if} \,\,\, d_{n,k}=[t^n]A(t) \left(g(t)\right)^k. \ee Therefore, the Riordan matrices from the sequences shown in the On-Line Encyclopedia of Integer Sequences (OLEIS) also present the coefficients of the corresponding classic Sheffer-type polynomials. For example, sequence \seqnum{A129652} $$ 1,1,1, 3, 2, 1, 13, 9, 3, 1, 73, 52, 18, 4, 1, \ldots $$ presents both Riordan array $(e^{x/(1-x)}, x)$ and the corresponding Sheffer-type polynomial sequence: \bns &&p_0(x)=1\\ &&p_1(x)=1+x\\ &&p_2(x)=3+2x+x^2/2!\\ &&p_3(x)=13+9x+3x^2/2!+x^3/3!\ldots \ens The row sums of Riordan array $(e^{x/(1-x)}, x)$, i.e., the polynomial values at $x=1$, are \seqnum{A052844}, while diagonal sums of the array are \seqnum{A129653} ({\it cf}. \cite{Sloane06}). Other examples, including, \seqnum{A000262}, \seqnum{A084358}, \seqnum{A133289}, etc., can be also found in \cite{Sloane06}. Suppose that $\Phi(t)$ is an analytic function of $t$ or a formal power series in $t$, say \be\label{eq:1.6} \Phi(t)=\sum^\infty_{k=0}c_kt^k, \qquad c_k=[t^k]\Phi(t), \ee where $c_k$ can be real or complex numbers. Then, formally we have a sum of general form \be\label{eq:1.7} \Phi(xE)f(0)=\sum^\infty_{k=0}c_kf(k)x^k. \ee In certain cases, $\Phi(\alpha +\beta)$ or $\Phi(\alpha\beta)$ can be decomposed into something having a power series in $\beta$ as a part. Accordingly the operator $\Phi(xE)=\Phi(x+x\Delta)=\Phi(xe^D)$ can be expressed as some power series involving operators $\Delta^k$ or $D^k$'s. Then it may be possible to compute the right-hand side of (\ref{eq:1.7}) by means of operator-series in $\Delta^k$ or $D^k$'s. This idea could be readily applied to various elementary functions $\Phi(t)$. Therefore, we can obtain various transformation formulas as well as series expansion formulas for the series of the form (\ref{eq:1.7}). It is well-known that the Eulerian fraction is a powerful tool to study the Eulerian polynomial, Euler function and its generalization, Jordan function ({\it cf.} \cite{Comtet74}). The classical Eulerian fraction can be expressed in the form \be\label{eq:1.8} \alpha_m(x)=\frac{A_m(x)}{(1-x)^{m+1}}\quad (x\neq1), \ee where $A_m(x)$ is the $m$th degree Eulerian polynomial of the form \be\label{eq:1.9} A_m(x)=\sum^m_{j=0} j!S(m,j)x^j(1-x)^{m-j}, \ee $S(m,j)$ being Stirling numbers of the second kind, i.e., $j!S(m,j)=\left[ \Delta^j t^m\right]_{t=0},$ which is also denoted by $\left\{\begin{array}{l}m\\j\end{array}\right\}.$ Evidently $\alpha_m(x)$ can be written in the form ({\it cf}. \cite{HHST05}) \be\label{eq:1.10} \alpha_m(x)=\sum^m_{j=0}\frac{j!S(m,j)x^j}{(1-x)^{j+1}}. \ee In order to express some new formulas for certain general types of power series, we need to introduce the following extension of Euler fraction, denoted by $\alpha_n(x, A(x), B(x), g(x))$, using the generalized Sheffer-type polynomials associated with analytic functions or power series $A(t)$, $B(t)$, and $g(t)$, which satisfy conditions in Definition \ref{def:1.0}. \be\label{eq:1.11} \alpha_n(x, A(x), B(x), g(x)):=\sum^n_{j=0}\sum^\infty_{\ell =j}j!{\ell \choose j}p_\ell (x) S(n,j), \ee where $p_\ell(x)$ is the generalized Sheffer-type polynomial of degree $\ell$ defined in Definition \ref{def:1.0}. In particular, if $A(x)=1$ and $g(x)=x$, then $p_\ell(x)=B^{(\ell)}(0)x^\ell/\ell!$, and the generalized Euler fraction is hence \be\label{eq:1.12} \alpha_n(x,1,B(x),x)=\sum^n_{j=0}S(n,j)B^{(j)}(x)x^j \ee because \[ \sum^\infty_{\ell =j}j!{\ell \choose j}p_\ell (x) = \sum^\infty_{\ell =j}j!{\ell \choose j}B^{(\ell)}(0)\frac{x^\ell} {\ell !}=B^{(j)}(x)x^j. \] Obviously, $\alpha_m(x)$ defined by (\ref{eq:1.8}) can be presented as \[ \alpha_m(x)=A_m(x, 1, (1-x)^{-1}, x). \] From (\ref{eq:1.11}), two kinds of generalized Eulerian fractions in terms of $A(x)=1$, $g(x)=x$, $B(x)=(1+x)^a$ and $B(x)=(1-x)^{-a -1}$, respectively, with real number $a$ as a parameter, can be introduced respectively, namely \be\label{eq:1.13} A_n(x,1,(1+x)^a,x )=\sum^n_{j=0}{a\choose j}\frac{j!S(n,j)x^j}{(1+x)^{j-a}} \ee for $x\not= -1$, and \be\label{eq:1.14} A_n(x,1,(1-x)^{-a-1},x)=\sum^n_{j=0}{a+j\choose j}\frac{j!S(n,j)x^j}{(1-x)^{a+j+1}} \ee for $x\not= 1$. These two generalized Eulerian fractions were given in \cite{HHS073}. Another extension of the classical Eulerian fraction is presented in \cite{HsuShiue99}. Two major transformation and expansion formulas and their applications will be displayed in next section, and the convergence of the series in the formulas is presented in \ref{sec:3}. \section{Series transformation-expansion formulas}\label{sec:2} \setcounter{equation}{0} \begin{theorem}\label{thm:2.1} Let $\{f(k)\}$ be a sequence of numbers (in ${\bR}$ or ${\bC}$), and let $h(t)$ be infinitely differentiable. Assume $A(t)$, $B(t)$, and $g(t)$ are analytic functions in a disk centered at the origin or power series defined as in Definition \ref{def:1.0} and $\{p_n(x)\}$ is the generalized Sheffer-type polynomial sequence associated with $A(x)$, $B(x)$, and $g(x)$, we have formally \bn\label{eq:2.1} &&\sum^\infty_{n=0}f(n)p_n(x)=\sum^\infty_{n=0}\Delta^nf(0)\left( \sum^\infty_{\ell =n}{\ell \choose n}p_\ell(x)\right)\\ &&\sum^\infty_{n=0}h(n)p_n(x)=\sum^\infty_{n=0}\frac{1}{n!}h^{(n)}(0)\alpha_n(x, A(x), B(x), g(x)), \label{eq:2.2} \en where $S(n,j)$ is the Stirling numbers of the second kind, and $\alpha_n(x, A(x),$ $B(x), g(x))=\sum^n_{j=0}\sum^\infty_{\ell =j}j!{\ell \choose j}p_\ell(x)S(n,j)$ is the generalized Eulerian fraction defined as in (\ref{eq:1.11}). In particular, if $g(t)=t$, then the transformation and expansion formulas (\ref{eq:2.1}) becomes to \bn\label{eq:2.3} &&\sum^\infty_{n=0}\frac{f(n)}{n!}\sum^n_{\ell =0}{n\choose \ell} A^{(n-\ell)}(0)B^{(\ell)}(0)x^\ell \nonumber\\ &=&\sum^\infty_{n=0}\frac{\Delta^nf(0)}{n!}\left(\sum^n_{\ell =0}{n\choose \ell}A^{(n-\ell)}(1)B^{(\ell)}(x)x^\ell\right). \en \end{theorem} \begin{proof} Applying the operator $A(E)B(xg(E))$ to $f(t)$ at $t=0$, where $E$ is the shift operator, we obtain the left-hand side of (\ref{eq:2.1}). On the other hand, we have \bns &&\left. A(E)B(xg(E))f(t) \right|_{t=0}=\left. A(1+\Delta)B(xg(1+\Delta))f(t) \right|_{t=0}\\ &=&\sum^\infty_{\ell=0}p_\ell(x)\left. (1+\Delta )^\ell f(t)\right|_{t=0}=\sum^\infty_{\ell=0} \sum^\ell_{n=0}{\ell\choose n}p_\ell(x)\Delta^nf(0), \ens which implies the double sum on the right-hand side of (\ref{eq:2.1}). Similarly, for the infinitely differentiable function $h(t)$, we can present \bns &&\left. A(E)B(xg(E))h(t) \right|_{t=0}=\left. A(e^D)B(xg(e^D))h(t) \right|_{t=0}\\ &=&\sum^\infty_{\ell=0}p_\ell (x) e^{\ell D} \left.h(t)\right|_{t=0} =\sum^\infty_{\ell=0} p_\ell (x) \sum^\infty_{n=0} \frac{\ell^n}{n!}h^{(n)}(0) \\ &=&\sum^\infty_{n=0}\left(\sum^\infty_{\ell =0}p_\ell(x)\ell^n\right)\frac{h^{(n)}(0)}{n!} \ens By applying (\ref{eq:2.1}) to the internal sum of the rightmost side of the above equation for $f(t)=t^k$ and noting $S(k,j)=\left( \left.\Delta^j t^k\right|_{t=0}\right)/j!$, we obtain \bns &&\left. A(E)B(xg(E))h(t) \right|_{t=0}\\ &=&\sum^\infty_{n=0}\left( \sum^\infty_{j=0}\sum^\infty_{\ell =j}{\ell \choose j}\left.p_\ell(x)\Delta^jt^n\right|_{t=0}\right) \frac{h^{(n)}(0)}{n!}\\ &=&\sum^\infty_{n=0} \left(\sum^n_{j=0} \sum^\infty_{\ell =j}{\ell \choose j}p_\ell(x)j! S(n,j)\right) \frac{h^{(n)}(0)}{n!}\\ &=&\sum^\infty_{n=0}\frac{h^{(n)}(0)}{n!} \alpha_n(x,A(x), B(x), g(x)). \ens This completes the proof of the theorem. If $g(t)=t$, then we have formally $p_n(x)=\sum^n_{\ell =0} A^{(n-\ell)}(0)B^{(\ell)}(0)x^\ell/ ((n-\ell)! \ell !)$ and \bns &&\left. A(E)B(xE)f(t)\right|_{t=0}\\ &=&\left. A(1+\Delta) B(x+x\Delta)f(t)\right|_{t=0}\\ &=&\sum^\infty_{n=0}\left( \sum^n_{\ell =0}\frac{A^{(n-\ell)}(1)}{(n-\ell)!}\frac{B^{(\ell)}(x)}{\ell !}(x\Delta)^\ell\right)\left.f(t)\right|_{t=0}, \ens which can be written as the double sum on the right-hand side of (\ref{eq:2.3}). \end{proof} \begin{remark} When $f(t)$ and $h(t)$ are polynomials, the right-hand sides of (\ref{eq:2.1}) and (\ref{eq:2.2}) are finite sums, which can be considered as the closed forms of the corresponding left-hand side series. For this reason, we call formulas (\ref{eq:2.1}) and (\ref{eq:2.2}) the series transformation and expansion (or transformation-expansion) formulas. Thus, for the $r$th degree polynomial $\phi(t)$, from (\ref{eq:2.1}) and (\ref{eq:2.2}) we have two expansion formulas, \bn\label{eq:2.4} &&\sum^\infty_{n=0}\phi(n)p_n(x)=\sum^r_{n=0}\Delta^n\phi(0)\left( \sum^\infty_{\ell =n}{\ell \choose n}p_\ell(x)\right)\\ &&\sum^\infty_{n=0}\phi(n)p_n(x)=\sum^r_{n=0}\frac{1}{n!} \phi^{(n)}(0)\alpha_n(x, A(x), B(x), g(x)), \label{eq:2.5} \en where the right-hand sides can be considered as the GF's of $\{ \phi(n)p_n(x)\}$. \end{remark} \begin{corollary}\label{cor:3.2} Let $\{\alpha_n(x, A(x), B(x), g(x))\}$ be the generalized Eulerian fraction sequence defined by (\ref{eq:1.11}). Then the exponential generating function of the sequence is $A(e^{t})B(xg(e^{t}))$. In particular, the exponential generating function of sequence $\{\alpha_n(x,1,B(x),x)\}$ is $B(xe^{t})$. \end{corollary} \begin{proof} The exponential GF of $\{\alpha_n(x, A(x), B(x), g(x))\}$ can be written as \bns &&\sum^\infty_{n=0}\alpha_n(x, A(x),B(x), g(x))\frac{t^n}{n!} \\ &=&\sum^\infty_{n=0}\sum^n_{j=0}S(n,j)\sum^\infty_{\ell =j}{\ell \choose j}p_\ell (x) \frac{t^n}{n!}\\ &=&\sum^\infty_{n=0}\left(\sum^\infty_{j=0}\frac{1}{j!}\left. \Delta^ju^n\right|_{u=0}\sum^\infty_{\ell =j}{\ell \choose j}p_\ell (x)\right)\frac{t^n}{n!}. \ens Applying formula (\ref{eq:2.1}) for $f(j)=j^n$ into the double sum in the above parentheses yields \bns &&\sum^\infty_{n=0}\alpha_n(x, A(x),B(x), g(x))\frac{t^n}{n!} \\ &=&\sum^\infty_{n=0}\left( \sum^\infty_{j=0}j^np_j(x)\right)\frac{t^n}{n!}\\ &=&\sum^\infty_{j=0}p_j(x)e^{jt} =A(e^{t})B(xg(e^{t})). \ens Here, the last step is due to Definition \ref{def:1.0}. \end{proof} We now give two special cases of Theorem~\ref{thm:2.1}. \begin{corollary}\label{cor:3.3} Let $\{ f(k)\}$ be a sequence of numbers (in ${\bR}$ or ${\bC}$), and let $B(t)$ and $g(t)$ be infinitely differentiable on $[0, \infty)$. Then we have formally \bn &&\sum^\infty_{n=0}f(n)B^{(n)}(0)\frac{x^n}{n!} =\sum^\infty_{n=0}\Delta^nf(0)B^{(n)}(x)\frac{x^n}{n!} \label{eq:2.6}\\ &&\sum^\infty_{n=0}g(n)B^{(n)}(0)\frac{x^n}{n!} =\sum^\infty_{n=0}\frac{g^{(n)}(0)}{n!}\alpha_n(x, 1, B(x), x),\label{eq:2.7} \en where $\alpha_n(x,1,B(x),x)$ is the generalized Eulerian fraction defined by (\ref{eq:1.12}). \end{corollary} \begin{proof} By setting $A(t)=1$ and $g(t)=t$ into transformation-expansion formulas (\ref{eq:2.1}) and (\ref{eq:2.2}), we obtain formally $p_\ell(x)=B^{(\ell)}(0)x^\ell/\ell !$. Thus, the modified formulas (\ref{eq:2.1}) and (\ref{eq:2.2}) are respectively (\ref{eq:2.6}) and (\ref{eq:2.7}). \end{proof} \begin{example} Setting respectively $B(t)=(1-t)^{-m-1}$ ($t\not= 1$) and $B(t)=(1+t)^m$ ($t\not= -1$) into (\ref{eq:2.6}) and (\ref{eq:2.7}) yield the transformation-expansion formulas \bn &&\sum^\infty_{k=0}{m+k\choose k}f(k)x^k=\sum^\infty_{k=0}{m+k\choose k}\frac{x^k}{(1-x)^{m+k+1}}\Delta^kf(0)\label{eq:2.8}\\ &&\sum^\infty_{k=0}{m+k\choose k}h(k)x^k=\sum^\infty_{k=0}\frac{ \alpha_k(x, 1, (1-x)^{-m-1}, x)}{k!}D^kh(0)\label{eq:2.9} \en and \bn &&\sum^\infty_{k=0}{m\choose k}f(k)x^k=\sum^\infty_{k=0}{m\choose k}\frac{x^k}{(1+x)^{k-m}}\Delta^kf(0)\label{eq:2.10}\\ &&\sum^\infty_{k=0}{m\choose k}h(k)x^k=\sum^\infty_{k=0}\frac{\alpha_k(x,1, (1+x)^m, x)}{k!}D^kh(0),\label{eq:2.11} \en respectively, where $\alpha_k(x, 1, (1-x)^{-m-1}, x)$ and $\alpha_k(x, 1, (1+x)^m, x)$ are defined in (\ref{eq:1.14}) and (\ref{eq:1.13}), respectively. By substituting $m=0$ in formulas (\ref{eq:2.8}) and (\ref{eq:2.9}) or applying transform $x\mapsto -x$ and staking $m=-1$ in formulas (\ref{eq:2.10}) and (\ref{eq:2.11}), we obtain \bn &&\sum^\infty_{k=0}f(k)x^k=\sum^\infty_{k=0}\frac{x^k}{(1-x)^{k+1}}\Delta^kf(0)\label{eq:2.12}\\ &&\sum^\infty_{k=0}h(k)x^k=\sum^\infty_{k=0}\frac{\alpha_k(x)}{k!}D^kh(0), \label{eq:2.13} \en where $\alpha_k(x)$ is defined by (\ref{eq:1.8}) and (\ref{eq:1.9}). (\ref{eq:2.12}) and (\ref{eq:2.13}) were shown as in \cite{HHST05}. And (\ref{eq:2.12}) is an extension of the following well-known Euler series transform that can be found by setting $x=-1$ into (\ref{eq:2.12}): \[ \sum^\infty_{n=0}(-1)^nf(n)=\sum^\infty_{n=0}\frac{(-1)^n}{2^{n+1}}\Delta^nf(0). \] By applying operator $E^m$ and multiplying $x^m$ on the both sides of formula (\ref{eq:2.8}), we obtain its alternative form as follows: \be\label{eq:2.14} \sum^\infty_{k=m}{k\choose m}f(k)x^k=\sum^\infty_{k=0}{m+k\choose m}\frac{x^{k+m}}{(1-x)^{m+k+1}}\Delta^kf(m) \ee \end{example} \begin{example} Let $\lambda$ and $\theta$ be any real numbers. The generalized falling factorial $(t+\lambda|\theta)_p$ is usually defined by \[ (t+\lambda|\theta)_p=\Pi^{p-1}_{j=0}(t+\lambda-j\theta),\,\, (p\geq 1),\,\, (t+\lambda|\theta)_0=1. \] It is known that Howard's degenerate weighted Stirling numbers ({\it cf.} \cite{Howard85}) may be defined by the finite differences of $(t+\lambda|\theta)_p$ at $t=0$: \[ S(p,k,\lambda|\theta):=\frac{1}{k!}\left[ \Delta^k(t+\lambda|\theta)_p\right]_{t=0}. \] Then, using (\ref{eq:2.14}), (\ref{eq:2.8}), and (\ref{eq:2.10}) with $f(t)=(t+\lambda|\theta)_p$, we get \bn &&\sum^\infty_{k=m}{k\choose m}(k+\lambda|\theta)_px^k=\sum^p_{k=0}{m+k\choose k}\frac{k!S(p,k,\lambda|\theta)x^{k+m}}{(1-x)^{m +k+1}},\label{eq:2.15}\\ &&\sum^\infty_{k=0}{m+k\choose k}(k+\lambda|\theta)_px^k=\sum^p_{k=0}{m+k\choose k}\frac{k!S(p,k,\lambda|\theta)x^k}{(1-x)^{m +k+1}},\label{eq:2.16}\\ &&\sum^\infty_{k=0}{m\choose k}(k+\lambda|\theta)_px^k=\sum^p_{k=0}{m\choose k}\frac{k!S(p,k,\lambda|\theta)x^k}{(1+x)^{k-m}}.\label{eq:2.17} \en The particular case of (\ref{eq:2.17}) with $x=1$, namely, \[ \sum^m_{k=0}{m\choose k}(k+\lambda|\theta)_p=\sum^p_{k=0}{m\choose k}2^{m-k}k!S(p,k,\lambda|\theta), \] was given in formula (35) of \cite{HsuShiue99}, and the particular case of (\ref{eq:2.16}) with $m =0$ was considered in \cite{HsuShiue01}. It is also obvious that the classical Euler's summation formula for the arithmetic-geometric series ({\it cf}. for example, \cite[Lemma 2.7]{GouldWetweerapong99}) is implied by (\ref{eq:2.16}) with $\lambda =\theta =0$ and $m =0$, or by (\ref{eq:2.17}) with $\lambda =\theta=0$, $m=-1$, $x\mapsto -x$. \end{example} Some other series transformation-expansion formulas can be constructed formally from the above formulas by using integration or differentiation. For instance, taking the integral on the bother sides of (\ref{eq:2.12}) we obtain \be\label{eq:2.18} \sum^\infty_{k=1}\frac{f(k)x^k}{k}=-f(0)\ln (1-x)+\sum^\infty_{k=1}\frac{1}{k}\left(\frac{x}{1-x}\right)^k\Delta^kf(0), \ee which can also be considered as a special case of (\ref{eq:2.6}) for $B(t)=-\ln (1-t)$. Using the substituting rule $t\mapsto D$ into (\ref{eq:1.6}) and applying the resulting operator to an infinitely differentiable function $f$ with the similar argument shown in Theorem \ref{thm:2.1}, we have \be\label{eq:2.19} \left. A(D)B(xg(D))f(t)\right|_{t=0}=\sum^\infty_{n=0}p_n(x)f^{(n)}(0). \ee We now specify $A$, $B$ and $g$ in (\ref{eq:2.19}) to establish the following corollary. \begin{corollary}\label{cor:3.4} If $(A(t), B(t), g(t))=(t/(e^t-1), e^t, t)$, $(2/(e^t+1), e^t, t)$, $(t/(\ln (t+1)), e^t, \ln (t+1))$, then from (\ref{eq:2.19}) we have \bn &&Df(x+y)=\sum^\infty_{n= 0}\phi_n(x)D^n\Delta f(y)\label{eq:2.20}\\ &&\sum^\infty_{n= 0}\left( -\frac{1}{2}\right)^n \Delta^nf(x)=\sum^\infty_{n=0}E_n(x)D^nf(0)\label{eq:2.21}\\ &&\Delta f(x+y)=\sum^\infty_{n= 0}\psi_n(x)\Delta^n Df(y),\label{eq:2.22} \en where $\phi(x)$ and $\psi(x)$ are Bernoulli polynomials of the first and second kind, respectively, and $E_n(x)$ are Euler polynomials. \end{corollary} \begin{proof} If $(A(t), B(t), g(t))=(t/(e^t-1), e^t, t)$, $(2/(e^t+1), e^t, t)$, $(t/(\ln (t+1)), e^t, \ln (t+1))$, then the corresponding Sheffer-type polynomials are $p_n(x)=\phi_n(x)$, $E_n(x)$, and $\psi_n(x)$, respectively ({\it cf}.\ \cite[pp.\ 250, 309, 279]{Jordan65}), and the corresponding operators on the left-hand side of (\ref{eq:2.19}) for the different $(A(t), B(t), g(t))$ become respectively $(De^{xD}/(e^D-1)=DE^x/\Delta$, \[ \frac{2 e^{xD}}{e^D+1} =\frac{2E^x}{\Delta +2}=\sum^\infty_{n=0} (-1)^n\left( \frac{\Delta}{2}\right)^n, \] and $\Delta (\Delta +1)^x/(\ln (\Delta +1))=\Delta E^x/D$. Hence, the proof of the theorem is complete. \end{proof} The results in Corollary \ref{cor:3.4} were given in \cite{HHS07} by using different treatment for each individual formula while our method described here can be considered as a uniform approach, which can be used to find more transformation and expansion formulas. It is obvious that for $x=0$, formulas (\ref{eq:2.20})-(\ref{eq:2.22}) are specified as \bn &&Df(y)=\sum^\infty_{n= 0}\frac{B^{(1)}_n}{n!} D^n\Delta f(y)\label{eq:2.23}\\ &&\sum^\infty_{n= 0}\left( -\frac{1}{2}\right)^n \Delta^nf(0)=\sum^\infty_{n=0}e_nD^nf(0)\label{eq:2.24}\\ &&\Delta f(y)=\sum^\infty_{n= 0}b_n\Delta^n Df(y),\label{eq:2.25} \en where $B^{(1)}_n=n! \phi_n(0)$ is the first order generalized Bernoulli number, and $e_n=E_n(0)$, and $b_n=\psi_n(0)$. \section{Convergence of the series transformation-expansions}\label{sec:3} \setcounter{equation}{0} As may be conceived, various formulas displayed in the list in Section \ref{sec:2} may be employed to construct some summation formulas with estimable remainders ({\it cf}. the proof of Theorem \ref{thm:3.3} below). In what follows convergence problems related to the series expansions in Section \ref{sec:2} will be investigated. We now establish convergence conditions for the series expansions in (\ref{eq:2.6}) and (\ref{eq:2.7}). Suppose that $\{ f(k)\}$ and $\{ h(k)\}$ are bounded sequences (say $|f(k)|\theta$ such that $\theta |x|<\rho |x|<1.$ Thus for large $k$ we have $\left|f(k)\right|^{\frac{1}{k}}<\rho.$ Consequently, \bns \left|\Delta^kf(0)\right|^{\frac{1}{k}}&\leq &\left( \sum^k_{j=0}{k\choose j}|f(j)|\right)^{\frac{1}{k}}<(2)^{\frac{1}{k}}\rho\to \rho \ens as $k\to \infty$. Therefore, for every $x\in (-1/\theta, 0)$ \bns &&\overline{\lim_{k\to \infty}}\left| \frac{1}{k}\left(\frac{x}{1-x}\right)^k\Delta^kf(0)\right|^{\frac{1}{k}}\\ &=&\overline{\lim_{k\to \infty}}\left| \frac{x}{1-x}\right|\left|\Delta^kf(0)\right|^{\frac{1}{k}}\leq \rho \left|\frac{x}{1-x}\right|< \rho |x|<1. \ens Hence, from the root test, the series expansion on the right-hand side of (\ref{eq:2.18}) is convergent. Similarly, the expansion on the right-hand side of (\ref{eq:2.12}) converges as well. This completes the proof of the theorem. \end{proof} To extend the convergence intervals of the series expansions in (\ref{eq:2.12}) and (\ref{eq:2.13}), we need more precise estimation as follows. \begin{theorem}\label{thm:3.3} Let $\{f(k)\}$ be a sequence of numbers (in ${\bR}$ or ${\bC}$), and let $\theta =\overline{\displaystyle \lim_{k\to \infty}} |f(k)|^{1/k}$. Then for any given $x$ with $x\not= 0$ we have the convergent expressions (\ref{eq:2.12}) and (\ref{eq:2.13}) provided that $|x|\theta <1$. \end{theorem} \begin{proof} As we mentioned in the proof of Theorem \ref{thm:3.2}, it is sufficient to show the convergence of (\ref{eq:2.12}). For this purpose, we now find a remainder of the expansion of (\ref{eq:2.12}) as follows. Formally, we have \bns &&(1-xE)^{-1}=(1-x-x\Delta)^{-1}\\ &=&(1-x)^{-1}\left( 1-\frac{x}{1-x}\Delta\right)^{-1}\\ &=&(1-x)^{-1}\left\{ \sum^{n-1}_{\ell =0}\left(\frac{x}{1-x}\right)^\ell \Delta^\ell +\frac{\left(\frac{x}{1-x}\Delta\right)^n}{1-\left(\frac{x}{1-x}\Delta\right)}\right\}\\ &=&\sum^{n-1}_{\ell =0}\frac{x^\ell}{(1-x)^{\ell +1}}\Delta^\ell +\left(\frac{x}{1-x}\right)^n\frac{\Delta^n}{1-xE}\\ &=& \sum^{n-1}_{\ell =0}\frac{x^\ell}{(1-x)^{\ell +1}}\Delta^\ell +\left(\frac{x}{1-x}\right)^n\sum^\infty_{\ell =0}x^\ell E^\ell \Delta^n. \ens Since $E^\ell \Delta^nf(0)=\Delta^nE^\ell f(0)=\Delta^nf(\ell)$, applying operator $(1-xE)^{-1}$ and the rightmost operator shown above to $f(t)|_{t=0}$, respectively, yields \bn &&\left.(1-xE)^{-1}f(t)\right|_{t=0}=\sum^\infty_{k=0}f(k)x^k \nonumber\\ &=&\sum^{n-1}_{k=0} \frac{x^k}{(1-x)^{k+1}}\Delta^kf(0)+\frac{x^n}{(1-x)^n}\sum^\infty_{\ell =0}x^\ell \Delta^nf(\ell).\label{eq:3.1} \en Since $|x|\theta <1$ ($x\not= 0$), the convergence of the series expansion on the left-hand side of (\ref{eq:2.12}) or (\ref{eq:3.1}) is obtain. To prove the convergence of the right-hand side of (\ref{eq:3.1}), i.e., the remainder form of (\ref{eq:2.12}), it is sufficient to show that $\sum^\infty_{\ell =0} x^\ell \Delta^nf(\ell)$ is absolutely convergent. Choose $\rho >\theta$ such that \[ \theta |x|<\rho |x| <1. \] Thus, for large $k$ we have $|f(k)|^{1/k}<\rho$, i.e., $|f(k)|<\rho^k$. Consequently we have, for large $\ell$ \bns \left|\Delta^nf(\ell)\right|^{1/\ell}&\leq& \left( \sum^n_{j=0}{n\choose j} |f(\ell+j)|\right)^{1/\ell}\leq \left(\sum^n_{j=0}{n\choose j}\rho^{\ell+j}\right)^{1/\ell}\\ &=&\rho(1+\rho)^{n/\ell}\to \rho \ens as $\ell \to \infty$. Thus \[ \overline{\displaystyle\lim_{\ell \to \infty}}\left|x^\ell \Delta^nf(\ell)\right|^{1/\ell}\leq \rho |x|<1, \] so that the series on the right-hand side of (\ref{eq:3.1}) or (\ref{eq:2.12}) is also convergent absolutely under the given conditions. \end{proof} A sequence $\{ a_n\}$ is called a null sequence if for any given positive number $\epsilon$, there exists and integer $N$ such that every $n>N$ implies $|a_n|<\epsilon$. \cite{Knopp71} ({\it cf}. Theorem 4 in Section 43) pointed out that a linear combination of $\{a_n\}$, denoted by $\{a'_n=\sum^n_{k=0}c_{n,k}a_k\}$, is also a null sequence if the coefficient set $\{ c_{n,k}\}_{0\leq k\leq n}$ ($n=0,1,2,\ldots$) satisfies the following two conditions: \begin{enumerate} \item[(i)] Every column contains a null sequence, i.e., for fixed $k\geq 0$, $c_{n,k}\to 0$ when $n\to \infty$. \item[(ii)] There exists a constant $K$ such that the sum $|a_{n,0}|+|a_{n,1}|+\cdots +|a_{n,n}| < K$ for every $n$. \end{enumerate} By using this claim of the null sequence, we can have the following convergence result of the series expansions in (\ref{eq:2.12}) and (\ref{eq:2.13}). \begin{theorem}\label{thm:3.4} Suppose that $\{f(n)\}$ is a given sequence of numbers (real or complex) such that $\sum^\infty_{n=0}f(n)x^n$ is convergent for every $x\in \Omega$ with $\Omega \cap (-\infty, 0)\not=\phi$. Then the series expressions on the right-hand sides of (\ref{eq:2.12}) and (\ref{eq:2.13}) converge for every $x\in \Omega \cap (-\infty, 0)$. \end{theorem} \begin{proof} We write the remainder of expression (\ref{eq:3.1}) as follows. \bns &&R_n:=\frac{x^n}{(1-x)^n}\sum^\infty_{\ell =0}x^\ell \Delta^nf(\ell)\\ &=&\frac{x^n}{(1-x)^n}\sum^\infty_{\ell =0}\sum^n_{j=0}(-1)^{n-j}x^\ell{n\choose j}f(j+\ell)\\ &=&\frac{(-x)^n}{(1-x)^n}\sum^n_{j=0}{n\choose j}\sum^\infty_{\ell =0}(-1)^{j}x^\ell{n\choose j}f(j+\ell)\\ &=&\frac{(-x)^n}{(1-x)^n}\sum^n_{j=0}(-x)^{-j}{n\choose j}\sum^\infty_{\ell =j}x^\ell f(\ell)=\frac{(-x)^n}{(1-x)^n}\sum^n_{j=0}(-x)^{-j}{n\choose j}x_j, \ens where $x_j=\sum^\infty_{\ell =j}x^\ell f(\ell)$ ($0\leq j\leq n$). Since $\sum^\infty_{\ell =0}x^\ell f(\ell)$ converges, $x_j$ is the term of a null sequence, applying the result on the linear combination of a null sequence shown above, we find the coefficients of $x_j$ in the linear combination of the rightmost sum, \[ c_{n,j}:=\frac{(-x)^n}{(1-x)^n}(-x)^{-j}{n\choose j} \] satisfy the following two conditions: (1) If $j$ is fixed, we have $c_{n,j}\to 0$ as $n\to \infty$ because \[ \left|c_{n,j}\right| =\frac{|x|^{n-j}}{(1-x)^n}{n\choose j} < \frac{n^j}{(1-x)^n} \] and $1/(1-x)<1$ for every $x\in \Omega \cap [-1,0)$; and \[ \left| a_{n,j}\right|=\frac{|x|^{n-j}}{(1-x)^n}{n\choose j} <\left(\frac{|x|}{1-x}\right)^nn^j \] and $|x/(1-x)|<1$ for every $x\in \Omega \cap (-\infty, -1)$. (2) For every $n$ and for every $x\in \Omega \cap (-\infty, 0)$ we have \[ \sum^n_{j=0}\left| a_{n,j}\right|=\frac{1}{(1-x)^n} \sum^n_{j=0}(-x)^{n-j}{n\choose j} =1. \] Therefore, Theorem 4 in Section 43 of \cite{Knopp71} shows that $R_n$ is also the term of a null sequence, so the series on the right-hand side of (\ref{eq:2.12}) converges for every $x\in \Omega \cap (-\infty, 0)$. In addition, the convergence of the right-hand series expansion of (\ref{eq:2.13}) is followed. \end{proof} We now discuss the convergence of the series expansions in (\ref{eq:2.20})-(\ref{eq:2.22}). Actually, we may sort the series transformation-expansion formulas associated with $(A(t), B(t), t)$ into two classes. The first class includes only either the sum $\sum \beta_k D^k f$ or the sum $\sum \gamma_k E^kf$ in the formulas such as (\ref{eq:2.20}) and (\ref{eq:2.22}). The second class includes the sums $\sum \beta_k D^k f$ and/or $\sum \gamma_k E^kf$ on both sides of the transformation-expansion formulas like (\ref{eq:2.21}). We may establish the following convergence theorem. \begin{theorem}\label{thm:3.5} For the first class series expansions associated with $\sum \beta_k$ $D^k f$ (or $\sum \gamma_k E^kf$) defined above, their absolute convergence are ensured if $\ov{\lim}_{k\to\infty}$ $\left|D^k f\right|^{1/k}<1$ (or $\ov{\lim}_{k\to\infty}\left|E^k f\right|^{1/k}<1$) and $\left|\beta_k\right|\leq1$ (or $\left|\gamma_k\right|\leq 1$). The second class series expansions defined above absolutely converge if $\ov{\lim}_{k\to\infty}\left|D^k f\right|^{1/k}<1$, $\left|\beta_k\right|\leq1$, and $\left|\gamma_k\right|\leq (1/(e-1))^k$. \end{theorem} \begin{proof} The first half of the theorem is easy to be verified by using the root test. To prove the second half, we need the following statement: If $f\in C^\infty$, then $\ov{\lim}_{k\to\infty}\left|D^kf(y)\right|^{1/k}0$ such that \[ \displaystyle \max_{0\leq t\leq 1}\left|{x+t\choose k}\right|< M k^{[|x|]}. \] Thus it follows that \[ \ov{\lim_{k\to \infty}}\left|\psi_k(x)\right|^{1/k}\leq \ov{\lim_{k\to \infty}}\left(\int^1_0\left|{x+t\choose k}\right|dt\right)^{1/k}\leq \ov{\lim_{k\to\infty}}\left(Mk^{[|x|]}\right)^{1/k}=1. \] This is a verification of (\ref{eq:3.8}), and corollary is proved. \end{proof} \begin{remark} The convergence conditions given in Theorem \ref{thm:3.2} can be improved by restricting $f$ and using similar techniques shown in \cite{HHST05}. \end{remark} \section{Acknowledgments} The author would like to thank the editor and the referee for their valuable suggestions and help. \begin{thebibliography}{99} \bibitem{Comtet74} L. Comtet, {\it Advanced Combinatorics}, Reidel, Dordrecht, 1974. \bibitem{Egorychev84} G. P. 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Roman, {\it The Umbral Calculus}, Academic Press, 1984. \bibitem{RomanRota78} S. Roman and G.-C. Rota, The umbral calculus, {\it Advances in Math.} {\bf 27} 1978, 95--188. \bibitem{Sloane06} N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at \href{http://www.research.att.com/~njas/sequences/}{\tt http://www.research.att.com/$\sim$njas/sequences/}, 2008. \bibitem{Sofo03} A. Sofo, {\it Computational Techniques for the Summation of Series}, Kluwer, 2003. \bibitem{Wilf94} H. S. Wilf, {\it Generatingfunctionology}, 2nd Edition, Academic Press, 1994. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 41A58; Secondary 41A80, 65B10, 05A15, 33C45, 39A70. \medskip \noindent \emph{Keywords: } Sheffer-type polynomials, symbolic operator, power series, transformation-expansion, generalized Eulerian fractions, Stirling number of the second kind. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000262}, \seqnum{A052844}, \seqnum{A084358}, \seqnum{A129652}, \seqnum{A129653}, and \seqnum{A133289}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received November 20 2007; revised version received April 21 2008. Published in {\it Journal of Integer Sequences}, July 5 2008. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}