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\begin{center}
\vskip 1cm{\LARGE\bf
Investigating Geometric and Exponential \\
\vskip .1in
Polynomials with Euler-Seidel Matrices
}
\vskip 1cm
\large
Ayhan Dil and Veli Kurt \\
Department of Mathematics \\
Akdeniz University \\
07058 Antalya \\
Turkey\\
\href{mailto:adil@akdeniz.edu.tr}{\tt adil@akdeniz.edu.tr}\\
\href{mailto:vkurt@akdeniz.edu.tr}{\tt vkurt@akdeniz.edu.tr}\\
\end{center}
\vskip .2 in
\begin{abstract}
In this paper we use the Euler-Seidel matrix method to obtain some
properties of geometric and exponential polynomials and numbers. Some new
results are obtained and some known results are reproved.
\end{abstract}
\section{Introduction}
This work is based on the Euler-Seidel matrix method \cite{Dumont}
which is related to algorithms, combinatorics and generating
functions. This method is quite useful to investigate properties of some
special numbers and polynomials.
In this paper we consider the Euler-Seidel matrix method for some
combinatorial numbers and polynomials. This method is relatively easier than
the most of combinatorial methods to investigate the structure of such
numbers and polynomials.
We obtain new properties of geometric (or Fubini) polynomials and numbers.
In addition, we use this method to find out some equations and recurrence
relations of exponential (or Bell) polynomials and numbers, and Stirling
numbers of the second kind. Although some results in this paper are known,
this method provides different proofs as well as new identities.
We first consider a given sequence $(a_{n})_{n\geq 0}$, and then determine
the Euler-Seidel matrix corresponding to this sequence is recursively by the
formulae
\begin{eqnarray}
a_{n}^{0} &=&a_{n}\text{,}\quad (n\geq 0)\text{,} \label{1} \\
a_{n}^{k} &=&a_{n}^{k-1}+a_{n+1}^{k-1}\text{,}\quad (n\geq 0,\,k\geq 1)
\notag
\end{eqnarray}%
where $a_{n}^{k}$ represents the $k$th row and $n$th column entry. From
relation $(\ref{1})$, it can be seen that the first row and the first column
can be transformed into each other via the well known binomial inverse pair
as,%
\begin{equation}
a_{0}^{n}=\sum_{k=0}^{n}\binom{n}{k}a_{k}^{0}\text{,} \label{2}
\end{equation}%
and%
\begin{equation}
a_{n}^{0}=\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}a_{0}^{k}\text{.} \label{2+}
\end{equation}
Euler \cite{Euler} deduced the following proposition.
\begin{proposition}[Euler]
Let
\begin{equation*}
a(t)=\sum_{n=0}^{\infty }a_{n}^{0}t^{n}
\end{equation*}%
be the generating function of the initial sequence $(a_{n}^{0})_{n\geq 0}$.
Then the generating function of the sequence $(a_{0}^{n})_{n\geq 0}$ is
\begin{equation}
\overline{a}(t)=\sum_{n=0}^{\infty }a_{0}^{n}t^{n}=\frac{1}{1-t}a\left(
\frac{t}{1-t}\right) \text{.} \label{3}
\end{equation}
\end{proposition}
A similar statement was proved by Seidel \cite{Seidel} with
respect to the exponential generating function.
\begin{proposition}[Seidel]
\label{P}Let
\begin{equation*}
A(t)=\sum_{n=0}^{\infty }a_{n}^{0}\frac{t^{n}}{n!}
\end{equation*}%
be the exponential generating function of the initial sequence $%
(a_{n}^{0})_{n\geq 0}$. Then the exponential generating function of the
sequence $(a_{0}^{n})_{n\geq 0}$ is
\begin{equation}
\overline{A}(t)=\sum_{n=0}^{\infty }a_{0}^{n}\frac{t^{n}}{n!}=e^{t}A(t)\text{%
.} \label{4}
\end{equation}
\end{proposition}
Dumont \cite{Dumont} presented several examples of
Euler-Seidel matrices, mainly using Bernoulli, Euler, Genocchi, exponential
(Bell) and tangent numbers. He also attempted to give a polynomial
extension of the Euler-Seidel matrix method. In \cite{Diletal}, Dil et al.
obtained some identities on Bernoulli and allied polynomials by introducing
polynomial extension of this method. Mez\H{o} and
Dil \cite{MD} gave a detailed
study of the harmonic and hyperharmonic numbers using the
Euler-Seidel matrix method. Moreover, some results on $r-$Stirling numbers and a new
characterization of the Fibonacci sequence have been presented. In \cite{DM}%
\ Dil and Mez\H{o}\ presented another algorithm which depends on a
recurrence relation and two initial sequences. Using the algorithm which is
symmetric respect to the rows and columns, they obtained some relations
between Lucas sequences and incomplete Lucas sequences. Hyperharmonic
numbers have been investigated as well.
\section{Definitions and Notation}
Now we give a summary about some special numbers and polynomials which we
will need later.
\textbf{Stirling numbers of the second kind.}
Stirling numbers of the second kind\ $%
\begin{Bmatrix}
n \\
k%
\end{Bmatrix}%
$ are defined by means of generating functions as follows \cite{AS,C}:
\begin{equation}
\sum_{n=0}^{\infty }%
\begin{Bmatrix}
n \\
k%
\end{Bmatrix}%
\frac{x^{n}}{n!}=\frac{\left( e^{x}-1\right) ^{k}}{k!}\text{.} \label{5}
\end{equation}
\subsection{Exponential polynomials and numbers}
Exponential polynomials (or single variable Bell polynomials) $\phi
_{n}\left( x\right) $ are defined by \cite{BL1, Ri, R}
\begin{equation}
\phi _{n}\left( x\right) :=\sum_{k=0}^{n}\QATOPD\{ \} {n}{k}x^{k}\text{.}
\label{8}
\end{equation}
We refer $\cite{B2}$\ for comprehensive information on the exponential
polynomials.
The first few exponential polynomials are:%
\begin{equation}
\begin{tabular}{|l|}
\hline
$\phi _{0}\left( x\right) =1\text{,}$ \\ \hline
$\phi _{1}\left( x\right) =x\text{,}$ \\ \hline
$\phi _{2}\left( x\right) =x+x^{2}\text{,}$ \\ \hline
$\phi _{3}\left( x\right) =x+3x^{2}+x^{3}\text{,}$ \\ \hline
$\phi _{4}\left( x\right) =x+7x^{2}+6x^{3}+x^{4}\text{.}$ \\ \hline
\end{tabular}%
\text{.} \label{L1}
\end{equation}
The exponential generating function of the exponential polynomials is given
by \cite{C}
\begin{equation}
\sum_{n=0}^{\infty }\phi _{n}\left( x\right) \frac{t^{n}}{n!}=e^{x\left(
e^{t}-1\right) }\text{.} \label{7}
\end{equation}
The well known exponential numbers (or Bell numbers) $\phi _{n}$ $\left(
\cite{BL2, C, CG, T}\right) $\ are obtained by setting $x=1$ in $\left( \ref%
{8}\right) $,$\,$i.e,%
\begin{equation}
\phi _{n}:=\phi _{n}\left( 1\right) =\sum_{k=0}^{n}\QATOPD\{ \} {n}{k}\text{.%
} \label{6}
\end{equation}%
The first few exponential numbers are:%
\begin{equation}
\phi _{0}=1\text{, }\phi _{1}=1\text{, }\phi _{2}=2\text{, \ }\phi _{3}=5%
\text{, \ }\phi _{4}=15\text{.} \label{L2}
\end{equation}
They form sequence \seqnum{A000110} in Sloane's
{\it Encyclopedia}.
Readers may also consult the lengthy bibliography of Gould \cite{GU},
where several papers and books are listed about the exponential
numbers.
The following recurrence relations that we reprove with the Euler-Seidel matrix
method hold for exponential polynomials \cite{R};
\begin{equation}
\phi _{n+1}\left( x\right) =x\left( \phi _{n}\left( x\right) +\phi
_{n}^{^{\prime }}\left( x\right) \right) \label{10}
\end{equation}%
and%
\begin{equation}
\phi _{n+1}\left( x\right) =x\sum_{k=0}^{n}\binom{n}{k}\phi _{k}\left(
x\right) \text{.} \label{11}
\end{equation}
\subsection{Geometric polynomials and numbers}
Geometric polynomials (also known as Fubini polynomials) are defined as
follows \cite{B}:
\begin{equation}
F_{n}\left( x\right) =\sum_{k=0}^{n}%
\begin{Bmatrix}
n \\
k%
\end{Bmatrix}%
k!x^{k}\text{.} \label{13}
\end{equation}%
By setting $x=1$ in $\left( \ref{13}\right) $ we obtain geometric numbers
(or preferential arrangement numbers, or Fubini numbers) $F_{n}$ $\left(
\cite{Da, Gr}\right) $ as%
\begin{equation}
F_{n}:=F_{n}\left( 1\right) =\sum_{k=0}^{n}%
\begin{Bmatrix}
n \\
k%
\end{Bmatrix}%
k!\text{.} \label{14}
\end{equation}
The exponential generating function of the geometric polynomials is given by
\cite{B}
\begin{equation}
\frac{1}{1-x\left( e^{t}-1\right) }=\sum_{n=0}^{\infty }F_{n}\left( x\right)
\frac{t^{n}}{n!}\text{.} \label{15}
\end{equation}
Let us give a short list of these polynomials and numbers as follows%
\begin{equation*}
\begin{tabular}{|l|}
\hline
$F_{0}\left( x\right) =1\text{,}$ \\ \hline
$F_{1}\left( x\right) =x\text{,}$ \\ \hline
$F_{2}\left( x\right) =x+2x^{2}\text{,}$ \\ \hline
$F_{3}\left( x\right) =x+6x^{2}+6x^{3}\text{,}$ \\ \hline
$F_{4}\left( x\right) =x+14x^{2}+36x^{3}+24x^{4}$, \\ \hline
\end{tabular}%
\end{equation*}%
and
\begin{equation*}
F_{0}=1\text{, \ }F_{1}=1\text{, \ }F_{2}=3\text{, \ }F_{3}=13\text{, \ }%
F_{4}=75\text{.}
\end{equation*}
They form sequence \seqnum{A000670} in Sloane's
{\it Encyclopedia}
Geometric and exponential polynomials are connected by the relation $\left(
\cite{B}\right) $%
\begin{equation}
F_{n}\left( x\right) =\int_{0}^{\infty }\phi _{n}\left( x\lambda \right)
e^{-\lambda }d\lambda \text{.} \label{16}
\end{equation}
Now we state our results.
\section{Results obtained by the matrix method}
Although we define Euler-Seidel matrices as matrices of numbers, we can also
consider entries of these matrices as polynomials \cite{Diletal}.
Thus the generating functions that we mention in the
statement of Seidel's proposition turn out to be two variables generating
functions. Therefore from now on when we consider these generating functions
as exponential generating functions of polynomials we use the notations $%
A\left( t,x\right) $ and $\overline{A}\left( t,x\right) $. Using these
notations relation $\left( \ref{4}\right) $ becomes%
\begin{equation}
\overline{A}\left( t,x\right) =e^{t}A\left( t,x\right) . \label{16'}
\end{equation}
\subsection{Results on geometric numbers and polynomials}
This part of our work contains some relations on the geometric numbers and
polynomials, most of which seems to be new.
\begin{proposition}
We have%
\begin{equation}
2F_{n}=\sum_{k=0}^{n}\binom{n}{k}F_{k} \label{27}
\end{equation}%
and%
\begin{equation}
F_{n}=2\sum_{k=0}^{n}\binom{n}{k}\left( -1\right) ^{n-k}F_{k} \label{28}
\end{equation}%
where $n\geq 1$.
\end{proposition}
\begin{proof}
Let us set the initial sequence of an Euler-Seidel matrix as the sequence of
geometric numbers, i.e., $\left( a_{n}^{0}\right) _{n\geq 0}=\left(
F_{n}\right) _{n\geq 0}$. Then we have%
\begin{equation*}
\left[
\begin{array}{cccccc}
1 & 1 & 3 & 13 & 75 & \cdots \\
2 & 4 & 16 & 88 & \cdots & \\
6 & 20 & 104 & \cdots & & \\
26 & 124 & \cdots & & & \\
150 & \cdots & & & & \\
\cdots & & & & &
\end{array}%
\right] \text{.}
\end{equation*}%
Observing the first row and the first column we see that $a_{0}^{n}=2F_{n}$,
$n\geq 1$. Firstly we need a proof of this fact. Proposition $\ref{P}$
enables us to write%
\begin{equation*}
\overline{A}\left( t\right) =\sum_{n=0}^{\infty }a_{0}^{n}\frac{t^{n}}{n!}=%
\frac{e^{t}}{2-e^{t}}=2\frac{1}{2-e^{t}}-1=\sum_{n=1}^{\infty }2F_{n}\frac{%
t^{n}}{n!}+1\text{.}
\end{equation*}%
Now comparison of the coefficients of the both sides gives $a_{0}^{n}=2F_{n}$
where $n\geq 1$. Using this result with equations $\left( \ref{2}\right) $
and $\left( \ref{2+}\right) $ we get $\left( \ref{27}\right) $ and $\left( %
\ref{28}\right) $.
\end{proof}
\begin{proposition}
\label{F1}Geometric polynomials satisfy the following recurrence relation%
\begin{equation}
F_{n}\left( x\right) =x\sum_{k=0}^{n-1}\binom{n}{k}F_{k}\left( x\right)
\text{.} \label{30}
\end{equation}
\end{proposition}
\begin{proof}
Let us set the initial sequence of an Euler-Seidel matrix as the sequence of
geometric polynomials, i.e. $\left( a_{n}^{0}\right) _{n\geq 0}=\left(
F_{n}\left( x\right) \right) _{n\geq 0}$. Thus we obtain from $\left( \ref%
{16'}\right) $%
\begin{equation*}
A\left( t,x\right) =\sum_{n=0}^{\infty }F_{n}\left( x\right) \frac{t^{n}}{n!}%
=\frac{1}{1-x\left( e^{t}-1\right) }
\end{equation*}%
and%
\begin{equation}
\overline{A}\left( t,x\right) =\frac{e^{t}}{1-x\left( e^{t}-1\right) }\text{.%
} \label{28'}
\end{equation}%
Then differentiation with respect to $t$ yields%
\begin{equation*}
\overline{A}\left( t,x\right) =\left[ \frac{1}{x}-\left( e^{t}-1\right) %
\right] \frac{d}{dt}A\left( t,x\right)
\end{equation*}%
which can equally well be written as%
\begin{equation*}
\overline{A}\left( t,x\right) =\sum_{n=0}^{\infty }\left[ \frac{%
F_{n+1}\left( x\right) }{x}+F_{n+1}\left( x\right) -\sum_{k=0}^{n}\binom{n}{k%
}F_{k+1}\left( x\right) \right] \frac{t^{n}}{n!}\text{.}
\end{equation*}%
Equating coefficients of $\frac{t^{n}}{n!}$\ in the preceding equation yields%
\begin{equation}
a_{0}^{n}=\frac{F_{n+1}\left( x\right) }{x}-\sum_{k=1}^{n}\binom{n}{k-1}%
F_{k}\left( x\right) \text{.} \label{29}
\end{equation}%
In view of $\left( \ref{2}\right) $\ equation $\left( \ref{29}\right) $\
shows the validity of $\left( \ref{30}\right) $.
\end{proof}
\begin{remark}
As a special case we get $\left( \ref{27}\right) $ by setting $x=1$ in $%
\left( \ref{30}\right) $.
\end{remark}
\begin{corollary}
\begin{equation}
F_{n+1}\left( x\right) =\frac{x}{1+x}\sum_{k=0}^{n}\binom{n}{k}\left[
F_{k}\left( x\right) +F_{k+1}\left( x\right) \right] \text{.} \label{31}
\end{equation}
\end{corollary}
\begin{proof}
With the aid of Proposition \ref{F1} we can write%
\begin{eqnarray}
F_{n+1}\left( x\right) &=&x\sum_{k=0}^{n}\binom{n}{k}F_{k}\left( x\right)
+x\sum_{k=1}^{n}\binom{n}{k-1}F_{k}\left( x\right) \notag \\
&=&x\sum_{k=0}^{n-1}\binom{n}{k}\left[ F_{k}\left( x\right) +F_{k+1}\left(
x\right) \right] +xF_{n}\left( x\right) \label{31+}
\end{eqnarray}%
Then $\left( \ref{31+}\right) $ yields equation $\left( \ref{31}\right) $.
\end{proof}
Now we give an important recurrence relation for the geometric polynomials.
\begin{proposition}
\label{F2} The following recurrence relation holds for the geometric
polynomials:%
\begin{equation}
F_{n+1}\left( x\right) =x\frac{d}{dx}\left[ F_{n}\left( x\right)
+xF_{n}\left( x\right) \right] \text{.} \label{32}
\end{equation}
\end{proposition}
\begin{proof}
We may use $\left( \ref{11}\right) $ and $\left( \ref{16}\right) $ to
conclude that%
\begin{equation*}
F_{n+1}\left( x\right) =x\sum_{k=0}^{n}\binom{n}{k}\int_{0}^{\infty }\phi
_{k}\left( x\lambda \right) \lambda e^{-\lambda }d\lambda .
\end{equation*}%
Using $\left( \ref{8}\right) $ this becomes%
\begin{equation*}
F_{n+1}\left( x\right) =x\sum_{k=0}^{n}\binom{n}{k}\sum_{r=0}^{k}%
\begin{Bmatrix}
k \\
r%
\end{Bmatrix}%
x^{r}\int_{0}^{\infty }\lambda ^{r+1}e^{-\lambda }d\lambda
\end{equation*}%
from which it follows that%
\begin{equation*}
F_{n+1}\left( x\right) =x\sum_{k=0}^{n}\binom{n}{k}\sum_{r=0}^{k}%
\begin{Bmatrix}
k \\
r%
\end{Bmatrix}%
\left( r+1\right) !x^{r}\text{.}
\end{equation*}%
This can equally well be written by means of derivative as%
\begin{equation*}
F_{n+1}\left( x\right) =x\frac{d}{dx}x\sum_{k=0}^{n}\binom{n}{k}F_{k}\left(
x\right) \text{.}
\end{equation*}%
Now equation $\left( \ref{30}\right) $\ permits us to write%
\begin{equation*}
F_{n+1}\left( x\right) =x\frac{d}{dx}\left[ F_{n}\left( x\right)
+xF_{n}\left( x\right) \right] .
\end{equation*}
\end{proof}
We have the following relation between the geometric polynomials and their
derivatives.
\begin{corollary}
\begin{equation}
\sum_{k=0}^{n}\binom{n}{k}xF_{k}^{\prime }\left( x\right) =\sum_{k=1}^{n}%
\binom{n}{k-1}F_{k}\left( x\right) \label{33}
\end{equation}
\end{corollary}
\begin{proof}
Combining results of Proposition $\ref{F1}$ and Proposition $\ref{F2}$ gives
$\left( \ref{33}\right) .$
\end{proof}
\section{Some other applications of the method}
In this section, using the Euler-Seidel matrix method, we are able to reprove
some known identities of Stirling numbers of second kind, exponential
numbers and polynomials. Using this method one can extend these results to
the other sequences.
\subsection{Applications to the Stirling numbers of the second kind}
Setting the initial sequence of an Euler-Seidel matrix as the sequence of
the Stirling numbers of the second kind, i.e., $a_{n}^{0}=%
\begin{Bmatrix}
n \\
m%
\end{Bmatrix}%
$ where $m$ is a fixed nonnegative integer, we get the exponential
generating function of the first row as%
\begin{equation*}
A\left( t\right) =\frac{\left( e^{t}-1\right) ^{m}}{m!}\text{.}
\end{equation*}%
We obtain from $\left( \ref{2}\right) $%
\begin{equation}
a_{0}^{n}=\sum_{k=0}^{n}\binom{n}{k}%
\begin{Bmatrix}
k \\
m%
\end{Bmatrix}%
\text{.} \label{17'}
\end{equation}%
Equation $\left( \ref{4}\right) $ yields%
\begin{equation*}
\overline{A}\left( t\right) =\sum_{n=0}^{\infty }a_{0}^{n}\frac{t^{n}}{n!}%
=e^{t}\frac{\left( e^{t}-1\right) ^{m}}{m!}
\end{equation*}%
which can equally be written as%
\begin{equation}
\overline{A}\left( t\right) =\frac{d}{dt}\frac{\left( e^{t}-1\right) ^{m+1}}{%
\left( m+1\right) !}\text{.} \label{17''}
\end{equation}%
Comparison of the coefficients of $t^{n}$ in equation $\left( \ref{17''}%
\right) $ and consideration $\left( \ref{17'}\right) $\ yield the following
result:
\begin{equation}
\sum_{k=0}^{n}\binom{n}{k}%
\begin{Bmatrix}
k \\
m%
\end{Bmatrix}%
=%
\begin{Bmatrix}
n+1 \\
m+1%
\end{Bmatrix}%
\text{.} \label{19}
\end{equation}%
Hence with the help of $\left( \ref{19}\right) $ and binomial inversion
formula $\left( \ref{2+}\right) $ we obtain%
\begin{equation}
\sum_{k=0}^{n}\binom{n}{k}\left( -1\right) ^{n-k}%
\begin{Bmatrix}
k+1 \\
m+1%
\end{Bmatrix}%
=%
\begin{Bmatrix}
n \\
m%
\end{Bmatrix}%
\text{.} \label{20}
\end{equation}
Relations $\left( \ref{19}\right) $\ and $\left( \ref{20}\right) $\ can be
found in \cite{GKP} respectively as the equations (6.15) and (6.17) on page
265.
\subsection{Applications to the exponential numbers and polynomials}
\subsubsection{Applications to the exponential numbers}
Let us construct an Euler-Seidel matrix with the initial sequence $\left(
a_{n}^{0}\right) _{n\geq 0}=\left( \phi _{n}\right) _{n\geq 0}$. Then we get
the following Euler-Seidel matrix%
\begin{equation*}
\left[
\begin{array}{ccccccc}
1 & 1 & 2 & 5 & 15 & 52 & \cdots \\
2 & 3 & 7 & 20 & \cdots & & \\
5 & 10 & 27 & \cdots & & & \\
15 & 37 & \cdots & & & & \\
52 & \cdots & & & & & \\
\cdots & & & & & &
\end{array}%
\right] \text{.}
\end{equation*}%
From this matrix we observe that $a_{0}^{n}=$ $\phi _{n+1}$. Now we prove
this observation using generating functions. Since $\left( a_{n}^{0}\right)
_{n\geq 0}=\left( \phi _{n}\right) _{n\geq 0}$ we have%
\begin{equation*}
A\left( t\right) =e^{e^{t}-1}\text{.}
\end{equation*}%
Equation $\left( \ref{4}\right) $ enables us to write%
\begin{equation}
\overline{A}\left( t\right) =e^{e^{t}+t-1}=\frac{d}{dt}\left(
e^{e^{t}-1}\right) =\sum_{n=0}^{\infty }\phi _{n+1}\frac{t^{n}}{n!}\text{.}
\label{20'}
\end{equation}%
Comparison of the coefficients of both sides in $\left( \ref{20'}\right) $
gives%
\begin{equation}
a_{0}^{n}=\phi _{n+1} \label{20+}
\end{equation}%
the desired result. From $\left( \ref{2}\right) $ and $\left( \ref{20+}%
\right) $ it follows that
\begin{equation}
\phi _{n+1}=\sum_{k=0}^{n}\binom{n}{k}\phi _{k}. \label{21}
\end{equation}%
Now considering $\left( \ref{2+}\right) $ and $\left( \ref{21}\right) $
together we obtain%
\begin{equation}
\phi _{n}=\sum_{k=0}^{n}\binom{n}{k}\left( -1\right) ^{n-k}\phi _{k+1}\text{.%
} \label{22}
\end{equation}
The identity $\left( \ref{21}\right) $ can be found in \cite{GKP} on page
373 and $\left( \ref{22}\right) $ is inverse binomial transform of the
identity $\left( \ref{21}\right) $.
\subsubsection{Applications to the exponential polynomials}
Setting the initial sequence of an Euler-Seidel matrix as the sequence of
exponential polynomials, i.e., $\left( a_{n}^{0}\right) _{n\geq 0}=\left(
\phi _{n}\left( x\right) \right) _{n\geq 0}$ we get following Euler-Seidel
matrix%
\begin{equation*}
\left[
\begin{array}{ccccc}
1 & x & x+x^{2} & x+3x^{2}+x^{3} & \cdots \\
1+x & 2x+x^{2} & 2x+4x^{2}+x^{3} & \cdots & \\
1+3x+x^{2} & 4x+5x^{2}+x^{3} & \cdots & & \\
1+7x+6x^{2}+x^{3} & \cdots & & & \\
\cdots & & & &
\end{array}%
\right] \text{.}
\end{equation*}%
We claim that $xa_{0}^{n}=\phi _{n+1}\left( x\right) $. Now we prove this
fact. With the aid of Proposition $\ref{P}$ we can write%
\begin{equation}
\overline{A}\left( t,x\right) =e^{t}e^{x\left( e^{t}-1\right) }=\frac{1}{x}%
\frac{d}{dt}e^{x\left( e^{t}-1\right) }\text{.} \label{22'}
\end{equation}%
Comparison of the coefficients of the both sides in $\left( \ref{22'}\right)
$ gives%
\begin{equation}
xa_{0}^{n}=\phi _{n+1}\left( x\right) \label{22+}
\end{equation}
the desired result. Hence equalities $\left( \ref{2}\right) $ and $\left( %
\ref{22+}\right) $ constitute a new proof of the equation $\left( \ref{11}%
\right) $.
Equation $\left( \ref{2+}\right) $ and equation $\left( \ref{11}\right) $
show the validity of the following equation%
\begin{equation}
x\phi _{n}\left( x\right) =\sum_{k=0}^{n}\binom{n}{k}\left( -1\right)
^{n-k}\phi _{k+1}\left( x\right) \text{.} \label{24}
\end{equation}
It is clear that equations $\left( \ref{11}\right) $ and $\left( \ref{24}%
\right) $ are the generalizations of equations $\left( \ref{21}\right) $ and
$\left( \ref{22}\right) ,$\ respectively.
With the help of generating functions technique we derive more results for
the exponential polynomials. The following remark is a new proof of the
equation $\left( \ref{10}\right) $.
\begin{remark}
Differentiation both sides of the equation $\left( \ref{7}\right) $ with
respect to $x$ we get%
\begin{equation*}
\sum_{n=0}^{\infty }\phi _{n}^{^{\prime }}\left( x\right) \frac{t^{n}}{n!}%
=e^{t}e^{x\left( e^{t}-1\right) }-e^{x\left( e^{t}-1\right) },
\end{equation*}%
which combines with $\left( \ref{22'}\right) $ to give%
\begin{equation*}
\sum_{n=0}^{\infty }\phi _{n}^{^{\prime }}\left( x\right) \frac{t^{n}}{n!}=%
\overline{A}\left( t,x\right) -A\left( t,x\right) \text{.}
\end{equation*}%
Then the last equation gives equation $\left( \ref{10}\right) $ by comparing
coefficients.
\end{remark}
Our last remark is about a relation between the exponential polynomials and
their derivatives.
\begin{remark}
Employing $\left( \ref{10}\right) $ in the equation $\left( \ref{24}\right) $
we obtain%
\begin{equation}
\sum_{k=0}^{n-1}\binom{n}{k}\left( -1\right) ^{k}\phi _{k}\left( x\right)
=\sum_{k=1}^{n}\binom{n}{k}\left( -1\right) ^{k-1}\phi _{k}^{^{\prime
}}\left( x\right) \text{.} \label{26}
\end{equation}
\end{remark}
\section{Acknowledgments}
We would like to thank the referees for their helpful comments on the
manuscript.
This research is supported by Akdeniz University Scientific Research Project
Unit.
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\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B73; Secondary 11B83.
\noindent \emph{Keywords:} Euler-Seidel matrices, Stirling numbers, exponential
numbers and polynomials, geometric numbers and polynomials.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A000110} and
\seqnum{A000670}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received July 12 2010;
revised version received October 2 2010; February 9 2011; March 25 2011.
Published in {\it Journal of Integer Sequences}, March 26 2011.
\bigskip
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\noindent
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