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\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} \\
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\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}


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\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 $((1-t)^{-p},
t/(t-1))$ ($p&gt;0$), 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
&amp;&amp;p_0(x)=1\\
&amp;&amp;p_1(x)=1+x\\
&amp;&amp;p_2(x)=3+2x+x^2/2!\\
&amp;&amp;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}
&amp;&amp;\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)\\
&amp;&amp;\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}
&amp;&amp;\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\\ &amp;=&amp;\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
&amp;&amp;\left. A(E)B(xg(E))f(t) \right|_{t=0}=\left. A(1+\Delta)B(xg(1+\Delta))f(t) \right|_{t=0}\\
&amp;=&amp;\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
&amp;&amp;\left. A(E)B(xg(E))h(t) \right|_{t=0}=\left. A(e^D)B(xg(e^D))h(t) \right|_{t=0}\\
&amp;=&amp;\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) \\ &amp;=&amp;\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
&amp;&amp;\left. A(E)B(xg(E))h(t) \right|_{t=0}\\ &amp;=&amp;\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!}\\
&amp;=&amp;\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!}\\
&amp;=&amp;\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
&amp;&amp;\left. A(E)B(xE)f(t)\right|_{t=0}\\
&amp;=&amp;\left. A(1+\Delta) B(x+x\Delta)f(t)\right|_{t=0}\\
&amp;=&amp;\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}
&amp;&amp;\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)\\
&amp;&amp;\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
&amp;&amp;\sum^\infty_{n=0}\alpha_n(x, A(x),B(x), g(x))\frac{t^n}{n!} \\
&amp;=&amp;\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!}\\
&amp;=&amp;\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 
&amp;&amp;\sum^\infty_{n=0}\alpha_n(x, A(x),B(x), g(x))\frac{t^n}{n!} \\
&amp;=&amp;\sum^\infty_{n=0}\left( \sum^\infty_{j=0}j^np_j(x)\right)\frac{t^n}{n!}\\
&amp;=&amp;\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
&amp;&amp;\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}\\
&amp;&amp;\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
&amp;&amp;\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}\\
&amp;&amp;\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
&amp;&amp;\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}\\
&amp;&amp;\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
&amp;&amp;\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}\\
&amp;&amp;\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
&amp;&amp;\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}\\
&amp;&amp;\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}\\
&amp;&amp;\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
&amp;&amp;Df(x+y)=\sum^\infty_{n= 0}\phi_n(x)D^n\Delta f(y)\label{eq:2.20}\\
&amp;&amp;\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}\\
&amp;&amp;\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
&amp;&amp;Df(y)=\sum^\infty_{n= 0}\frac{B^{(1)}_n}{n!} D^n\Delta f(y)\label{eq:2.23}\\
&amp;&amp;\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}\\
&amp;&amp;\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 $|f(k)|&lt;M$ and $|h(k)|&lt;M$ for all $k$), and that $g(z)$ is analytic for $|z|&lt;\rho$. Then it is follows that the left-hand sides of (\ref{eq:2.6}) and (\ref{eq:2.7}) as well as the right-hand sides of (\ref{eq:2.6}) and (\ref{eq:2.7}) are absolutely convergent series for $|x|&lt;\rho$. 
Hence, we have the following convergence theorem. 

\begin{theorem}\label{thm:3.1}
If $\{ f(k)\}$ and $\{ g(k)\}$ are bounded sequences, and that $B(z)$ is analytic for $|z|&lt;\rho$ for some positive real number $\rho$, 
then the series expansions in (\ref{eq:2.6}) and (\ref{eq:2.7}) converge absolutely for all $|x|&lt;\rho$.  
\end{theorem}


The convergence on the general case where $\{ f(k)\}$ is not bounded
presents some complicated situation. The next theorem gives a
discussion for the series transformation-expansion formulas shown in
(\ref{eq:2.12}), (\ref{eq:2.13}), and (\ref{eq:2.18}), and general way
may be developed through it, which is left for the interested reader to
consider.


\begin{theorem}\label{thm:3.2}
Let $\{f(k)\}$ be a sequence of numbers (in ${\bR}$ or ${\bC}$), and denote $\theta:=\overline{\lim}_{k\to \infty}\left|f(k)\right|^{1/k}$. 
Then the series expansions in (\ref{eq:2.12}),  (\ref{eq:2.13}), and (\ref{eq:2.18}) are convergent for all nonzero $x$ satisfying $|x|\theta&lt;1$. 
\end{theorem}

\begin{proof}
Substituting the expression of $\alpha_k(x)$ defined by (\ref{eq:1.10}) into (\ref{eq:2.12}) and noting 
$j!\sum^\infty_{k=j} S(k,j)\frac{D^k}{k!}=(e^D-1)^j=\Delta^j$ ({\em cf.} \cite{HHST05}) yields (\ref{eq:2.13}). More precisely, 

\bns
&amp;&amp;\sum^\infty_{k=0}g(k)x^k =\sum^\infty_{k=0} \frac{\alpha_k(x)}{k!}D^kg(0)\\
&amp;=&amp; \sum^\infty_{k=0}\sum^k_{j=0}\frac{j!}{k!}S(k,j)\frac{x^j}{(1-x)^{j+1}}D^kg(0)\\
&amp;=&amp; \sum^\infty_{j=0}\frac{x^j}{(1-x)^{j+1}}\left(j!\sum^\infty_{k=j} 
S(k,j)\frac{D^k}{k!}\right)g(0)\\
&amp;=&amp;\sum^\infty_{j=0}\frac{x^j}{(1-x)^{j+1}}\Delta^jg(0).
\ens
Hence, we only need to show the convergence of expansions in (\ref{eq:2.12}) and (\ref{eq:2.18}). 

In accordance with Cauchy's root test, the convergence of the series on
the left-hand side of (\ref{eq:2.12}) and (\ref{eq:2.18}) is obvious
because of the condition $|x|\theta &lt;1$. To prove the convergence of
the series expansion on the right-hand side of (\ref{eq:2.12}), we
choose $\rho &gt;\theta$ such that $\theta |x|&lt;\rho |x|&lt;1.$  Thus for
large $k$ we have $\left|f(k)\right|^{\frac{1}{k}}&lt;\rho.$ Consequently,

\bns
\left|\Delta^kf(0)\right|^{\frac{1}{k}}&amp;\leq &amp;\left( \sum^k_{j=0}{k\choose j}|f(j)|\right)^{\frac{1}{k}}&lt;(2)^{\frac{1}{k}}\rho\to \rho
\ens
as $k\to \infty$. Therefore, for every $x\in (-1/\theta, 0)$ 

\bns
&amp;&amp;\overline{\lim_{k\to \infty}}\left| \frac{1}{k}\left(\frac{x}{1-x}\right)^k\Delta^kf(0)\right|^{\frac{1}{k}}\\
&amp;=&amp;\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|&lt; \rho |x|&lt;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 $|x|\theta &lt;1$. 
\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
&amp;&amp;(1-xE)^{-1}=(1-x-x\Delta)^{-1}\\
&amp;=&amp;(1-x)^{-1}\left( 1-\frac{x}{1-x}\Delta\right)^{-1}\\
&amp;=&amp;(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\}\\
&amp;=&amp;\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}\\
&amp;=&amp; \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
&amp;&amp;\left.(1-xE)^{-1}f(t)\right|_{t=0}=\sum^\infty_{k=0}f(k)x^k
\nonumber\\
&amp;=&amp;\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 $|x|\theta &lt;1$ ($x\not= 0$), 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 $\rho &gt;\theta$ such that

\[
\theta |x|&lt;\rho |x| &lt;1.
\] 
Thus, for large $k$ we have $|f(k)|^{1/k}&lt;\rho$, i.e., $|f(k)|&lt;\rho^k$. Consequently we have, for large $\ell$ 

\bns
\left|\Delta^nf(\ell)\right|^{1/\ell}&amp;\leq&amp; \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}\\
&amp;=&amp;\rho(1+\rho)^{n/\ell}\to \rho
\ens
as $\ell \to \infty$. Thus 

\[
\overline{\displaystyle\lim_{\ell \to \infty}}\left|x^\ell \Delta^nf(\ell)\right|^{1/\ell}\leq \rho |x|&lt;1,
\]
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 $n&gt;N$ implies $|a_n|&lt;\epsilon$. \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 $|a_{n,0}|+|a_{n,1}|+\cdots +|a_{n,n}| &lt; K$ 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
&amp;&amp;R_n:=\frac{x^n}{(1-x)^n}\sum^\infty_{\ell =0}x^\ell \Delta^nf(\ell)\\
&amp;=&amp;\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)\\
&amp;=&amp;\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)\\
&amp;=&amp;\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   

\[
\left|c_{n,j}\right| =\frac{|x|^{n-j}}{(1-x)^n}{n\choose j}
&lt; \frac{n^j}{(1-x)^n}
\]
and $1/(1-x)&lt;1$ for every $x\in \Omega \cap [-1,0)$; and 

\[
\left| a_{n,j}\right|=\frac{|x|^{n-j}}{(1-x)^n}{n\choose j}
&lt;\left(\frac{|x|}{1-x}\right)^nn^j
\]
and $|x/(1-x)|&lt;1$ 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 $\ov{\lim}_{k\to\infty}$ $\left|D^k f\right|^{1/k}&lt;1$ (or $\ov{\lim}_{k\to\infty}\left|E^k f\right|^{1/k}&lt;1$) and 
$\left|\beta_k\right|\leq1$ (or $\left|\gamma_k\right|\leq 1$). 

The second class series expansions defined above  
absolutely converge if $\ov{\lim}_{k\to\infty}\left|D^k f\right|^{1/k}&lt;1$, $\left|\beta_k\right|\leq1$, 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  $\ov{\lim}_{k\to\infty}\left|D^kf(y)\right|^{1/k}&lt;a$, a positive real number,  implies  

\be\label{eq:3.2}
\ov{\lim_{k\to\infty}}\left|\Delta^kf(y)\right|^{1/k}&lt;e^a-1.
\ee
In fact, if we denote $\ov{\lim}_{n\to\infty}$
$\left|D^nf(y)\right|^{1/n}=\theta$, then there exists a number
$\gamma$ such that $\theta &lt;\gamma &lt; a$. Thus for large enough $n$ we
have $\left| D^nf(y)\right|^{1/n}&lt;\gamma$ or
$\left|D^nf(y)\right|&lt;\gamma^n$.

Noting $S(n,m)\geq 0$ and 
$\left|D^nf(y)\right|&lt;\gamma^n$, we obtain  

\bns
\left| \Delta^kf(y)\right|&amp;=&amp;\left| \sum_{n\geq k}\frac{k!}{n!}S(n,k)
D^n f(y) \right| \leq \sum_{n\geq k}\frac{k!}{n!}S(n,k)\left| D^nf(y)\right|\\
&amp;\leq&amp; \sum_{n\geq k}\frac{k!}{n!}S(n,k)\gamma^n
= (e^\gamma -1)^k&lt;(e^a-1)^k.
\ens
Here the rightmost equality is from Jordan \cite[p.\ 176]{Jordan65}.

Therefore, $\ov{\lim}_{k\to\infty}\left|D^k f\right|^{1/k}&lt;1$ implies
that $\ov{\lim_{k\to\infty}}\left|\Delta^kf(y)\right|^{1/k}&lt;e-1$. Those
two inequalities and the conditions for the coefficients $\{ \beta_k\}$
and $\{\gamma_k\}$ confirm the absolute convergence of the second class
series expansions with the root test.
\end{proof}

As a corollary of Theorem \ref{thm:3.5}, we now establish the convergence results of the series expansions in (\ref{eq:2.20})-(\ref{eq:2.22}).
 
\begin{corollary}\label{cor:3.6}
For given $f\in C^\infty$ and $x,y\in {\bR}$, the absolute convergence of the series expansion (\ref{eq:2.20}) is ensured by the condition

\be\label{eq:3.3}
\ov{\lim_{k\to \infty}}\left|\Delta D^kf(y)\right|^{1/k}&lt;1.
\ee

Similarly, the absolute convergence of the series expansion (\ref{eq:2.21}) and (\ref{eq:2.22}) are ensured by the conditions

\be\label{eq:3.4}
\ov{\lim_{k\to\infty}}\left|\Delta^kDf(y)\right|^{1/k}&lt;1
\ee
and

\be\label{eq:3.5}
\ov{\lim_{k\to\infty}}\left|D^kf(y)\right|^{1/k}&lt;1,
\ee
respectively. 
\end{corollary}

\begin{proof}
From Theorem \ref{thm:3.5}, it is sufficient to show that 

\be\label{eq:3.6}
\ov{\lim_{k\to \infty}}\left|\phi_k(x)\right|^{1/k}\leq 1,
\ee
\be\label{eq:3.7}
\ov{\lim_{k\to \infty}}\left|E_k(x)\right|^{1/k}\leq 1,
\ee
and

\be\label{eq:3.8}
\ov{\lim_{k\to \infty}}\left|\psi_k(x)\right|^{1/k}\leq 1,
\ee
which will be proved below from the basic properties of $\phi_k(x)$, $E_k(x)$, and $\psi_k(x)$ shown as in Jordan \cite{Jordan65}.

Write the Bernoulli polynomials of the first kind, $\phi_k (x)$, as
({\it cf.} Jordan \cite[p.\ 321]{Jordan65})

\[
\phi_k(x)=\sum^k_{j=0}\frac{x^{k-j}}{(k-j)!}\alpha_j,
\]
where $\alpha_j=B^{(1)}_j/j!$, and $B^{(1)}_j$ are ordinary Bernoulli numbers. Note that $\alpha_0=1$, $\alpha_1=-1/2$, $\alpha_{2m+1}=0$ ($m\in {\bN}$) and ({\it cf.} \cite[p.\ 245]{Jordan65})

\[
\left| \alpha_{2m}\right|\leq \frac{1}{12(2\pi)^{2m-2}}, \,\, 
(m=0,1,2,\ldots).
\]
Thus for $k\geq 2$ 

\bns
\left|\phi_k(x)\right|&amp;\leq &amp;\frac{|x|^k}{k!}+\frac{|x|^{k-1}}{2(k-1)!}
+\sum^k_{j=2}\left(\frac{1}{12(2\pi)^{j-2}}\right)\frac{|x^{k-j}|}{(k-j)!}\\
&amp;&lt;&amp; \sum^k_{j=0}\frac{|x|^j}{j!}\leq e^{|x|}.
\ens
It follows that $|\phi_k(x)|^{1/k}&lt;exp(|x|/k)\to 1$ as $k\to \infty$, which implies (\ref{eq:3.6}).

Secondly, note that Euler polynomial 
$E_k(x)$ can be written in the form

\be\label{eq:3.9}
E_k(x)=\sum^k_{j=0}e_j\frac{x^{k-j}}{(k-j)!}, \,\, (e_0=1),
\ee
where $e_j=E_j(0)$, $e_{2m}=0$ ($m=1,2,\ldots$), and $e_{2m-1}$ satisfies the inequality ({\it cf.} \cite[p.\ 302]{Jordan65}).
                                                                                                              
\be\label{eq:3.10}
\left|e_{2m-1}\right|&lt; \frac{2}{3\pi^{2m-2}}&lt;1\,\, (m=1,2,\ldots).
\ee
Thus we have the estimation
			 
\[
\left| E_k(x)\right|\leq \frac{|x|^k}{k!}+\sum^k_{j=1}|e_j|\frac{|x|^{k-j}}{(k-j)!}\leq   \frac{|x|^k}{k!}+\sum^k_{j=1} \frac{|x|^{k-j}}{(k-j)!}&lt;e^{|x|}. 
\]                                              
Consequently we get

\[
\ov{\lim_{k\to\infty}}\left| E_k(x)\right|^{1/k}\leq \lim_{k\to\infty}\left(e^{|x|}\right)^{1/k}=1. 
\]                                    
Hence (\ref{eq:3.7}) is verified. 

Finally, from \cite[p.\ 268]{Jordan65}, we have an  integral
representation of  $\psi_k(x)$,  the Bernoulli polynomials of the
second kind, namely

\be\label{eq:3.11}                                                          
\psi_k(x)=\int^1_0{x+t\choose k}dt.
\ee
For $t\in [0,1]$ and for large $k$ we have the estimation
 
\[
\left|{x+t\choose k}\right|=\frac{\left|(x+t)_k\right|}{k!} 
=\frac{\left|(k-x-t-1)_k\right|}{k!}=o\left(\frac{k+[|x|])_k}{k!}\right)
=o\left(k^{[|x|]}\right).
\]
This means that there is a constant $M&gt;0$ such that 

\[
\displaystyle \max_{0\leq t\leq 1}\left|{x+t\choose k}\right|&lt; M k^{[|x|]}.
\]
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. Egorychev, {\it Integral Representation and the 
Computation of Combinatorial Sums},
Translation of Math. Monographs, Vol.\ 59, American Mathematical
Society, 1984. 

\bibitem{GouldWetweerapong99}
H. W. Gould and J. Wetweerapong, Evaluation of some classes of binomial
identities and two new sets of polynomials, {\it Indian J. Math.} {\bf 41}
(1999), 159--190.

\bibitem{HHST05}
T. X. He, L. C. Hsu, P. J.-S. Shiue, and D. C. Torney, A symbolic
operator approach to several summation formulas for power series, {\it J.
Comp. Appl. Math.} {\bf 177} (2005), 17--33.

\bibitem{HHS073}
T. X. He, L. C. Hsu, and P. J.-S. Shiue, A symbolic operator approach
to several summation formulas for power series II,  2007.

\bibitem{HHS072}
T. X. He, L. C. Hsu, and P. J.-S. Shiue, The Sheffer group and the
Riordan group, {\it Discrete Appl. Math.} {\bf 155} (2007), 1895--1909.

\bibitem{HHS07}
T. X. He, L. C. Hsu, and P. J.-S. Shiue,
Symbolization of generating functions, an application of Mullin-Rota's
theory of binomial enumeration, {\it Comp. Math. Appl.} {\bf 54}
(2007), 664--678.

\bibitem{Howard85}
F. T. Howard, Degenerate weighted Stirling numbers, {\it Discrete Math.}
{\bf 57} (1985), 45--58.

\bibitem{HsuShiue01}	
L. C. Hsu and P. J.-S. Shiue, Cycle indicators and special functions,
{\it Ann. Combinatorics} {\bf 5} (2001), 179--196.

\bibitem{HsuShiue99}
L. C. Hsu and P. J.-S. Shiue, On certain summation problems and
generalizations of  Eulerian polynomials and numbers, {\it Discrete
Math.} {\bf 204} (1999), 237--247.

\bibitem{Jordan65}
Ch. Jordan, {\it Calculus of Finite Differences}, Chelsea, New York, 1965.

\bibitem{Knopp71}
K. Knopp, {\it Theory and Application of Infinite Series}, Hafner Publishing
Comp., New York, 1971.

\bibitem{Petkovsek96}
M. Petkov\v{s}ek, H. S. Wilf, and D. Zeilberger, {\it A=B}, AK Peters, 
1996.

\bibitem{Roman84}
S. 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}






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\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
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\vspace*{+.1in}
\noindent
Received November 20 2007;
revised version received   April 21 2008.
Published in {\it Journal of Integer Sequences}, July 5 2008.

\bigskip
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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\end{document}

                                                                                

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