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\begin{center}
\vskip 1cm{\LARGE\bf An Identity for Generalized \\
\vskip .02in
Bernoulli Polynomials\\
}
\vskip 1cm
\large
Redha Chellal\footnote{Corresponding author.} and Farid Bencherif\\
LA3C, Faculty of Mathematics\\
USTHB\\
Algiers\\
Algeria\\
\href{mailto:rchellal@usthb.dz}{\tt rchellal@usthb.dz}\\
\href{mailto:chellalredha4@gmail.com}{\tt chellalredha4@gmail.com}\\
\href{mailto:fbencherif@usthb.dz}{\tt fbencherif@usthb.dz}\\
\ \\
Mohamed Mehbali\\
Centre for Research Informed Teaching \\
London South Bank University\\
London\\
United Kingdom\\
\href{mailto:mehbalim@lsbu.ac.uk}{\tt mehbalim@lsbu.ac.uk}\\
\end{center}

\vskip .2 in

\begin{abstract}
Recognizing the great importance of Bernoulli numbers and Bernoulli
polynomials in various branches of mathematics, the present paper
develops two results dealing with these objects.
The first one proposes an identity for the
generalized Bernoulli polynomials, which leads to further generalizations
for several relations involving classical Bernoulli numbers and Bernoulli
polynomials. In particular, it generalizes a recent identity suggested by
Gessel. The second result allows the deduction of similar identities for
Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized 
Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling.
\end{abstract}

\section{Introduction}\label{sec:intro}

Let $\mathbb{N}$ and $\mathbb{C}$ denote, respectively, the set of positive integers and the set of complex numbers. In his book, Roman \cite[p.\ 93]{rom} defined generalized Bernoulli polynomials
$B_{n}^{(\alpha)}(x)$ as follows: for all $n \in \mathbb{N}$ and $\alpha \in \mathbb{C}$, we have
\begin{equation}
\sum_{n=0}^{\infty}B_{n}^{(\alpha)}(x)\frac{t^{n}
}{n!}=\bigg(\frac{t}{e^{t}-1}\bigg)^{\alpha}e^{tx}.  \label{cn1}
\end{equation}
The Bernoulli numbers $B_{n}$,
classical Bernoulli polynomials $B_{n}(x)$,
and generalized Bernoulli numbers $B_{n}^{(\alpha)}$ are,
respectively, defined by
\begin{equation}
B_{n}=B_{n}(0), \ B_{n}(x)=B_{n}^{(1)}(x), 
\text{ and } B_{n}^{(\alpha)}=B_{n}^{(\alpha)}(0).  \label{t2}
\end{equation}
The Bernoulli numbers and the Bernoulli polynomials play a fundamental role in various branches of mathematics, such as combinatorics, number theory,
mathematical analysis, and topology. Dilcher and Slavutskii \cite{dil} listed numerous publications on the properties of Bernoulli numbers and Bernoulli polynomials. 

In this paper, we are mainly interested in evaluating the sum
$S_{n,\ell,r}^{(\alpha)}(x,y,z)$ defined by
\begin{align}
S_{n,\ell,r}^{(\alpha)}(x,y,z)&amp;=\sum_{k=0}^{n+r}
\binom{n+r}{k}\binom{\ell+k+r}{r}x^{n+r-k}B_{\ell+k}^{(\alpha)}(y) \notag \\
&amp;+(-1)^{\ell+n+r+1}\sum_{k=0}^{\ell+r}\binom{\ell+r}{k}\binom{n+k+r}{r}
x^{\ell+r-k}B_{n+k}^{(\alpha)}(z), \label{t1}
\end{align}
where $n$, $\ell$, and $r$ are non-negative integers.

In 2011, Zekiri and Bencherif \cite{zek3} proved, for $r$ odd, that
\begin{equation}
S_{n,n,r}^{(1)}(1,0,0)=0.   \label{b22}
\end{equation}
In 2012, Bencherif and Garici \cite{ben1} improved Eq.~(\ref{b22}), for all non-negative integers $\ell$, and showed that
\begin{equation}
S_{n,\ell,r}^{(1)}(1,0,0)=0. \label{t3}
\end{equation}

In 2013, using umbral calculus, Gessel \cite[Thm.\ 2, p.\ 6]{bel}
generalized the result above by proving the following explicit formula
for the sum $S_{n,\ell,r}^{(1)}(m,0,0)$: \begin{equation} S_{n,\ell
,r}^{(1)}(m,0,0)=( r+1) \sum_{k=1}^{m-1}\sum_{j=0}^{r+1}(-1)^{\ell
+j-1}\binom{n+r}{j} \binom{\ell +r}{r+1-j}k^{\ell +j-1}(m-k)^{n+r-j},
\label{t4} \end{equation} for all integers $m\geq 1$.

By considering the special case where $\ell =n$ and $r$ is odd,
from Eq.~(\ref{t4}), Gessel \cite{bel} deduced the following formula:
\begin{align}
\frac{1}{2}S_{n,n,r}^{(1)}(m,0,0)&amp;=\sum_{k=0}^{n+r}m^{n+r-k}\binom{n+r}{k}
\binom{n+k+r}{r}B_{n+k}  \notag \\
&amp;=\frac{1}{2}(r+1)\sum_{k=1}^{m-1}\sum_{j=0}^{r+1}\binom{n+r}{j}
\binom{n+r}{r+1-j}k^{j+n-1}(k-m)^{n+r-j}. \label{t5}
\end{align}
The Relations (\ref{t3}), (\ref{t4}), and (\ref{t5}) are, in fact, the
synthesis of a long journey of research discussed in the third section.

This paper briefly reviews some properties of generalized
Bernoulli polynomials, as well as two related lemmas that help
to prove the result presented in the second section. The main
result proposes a simplified and useful expression for the sum
$S_{n,l,r}^{\left(\alpha\right)}\left(x,y,z\right)$ where $x+y+z-\alpha$
is a non-negative integer. The result is stated and proven in the fourth
section. Then the fifth section provides some applications related to
the main theorem. In the last section, we establish similar identities
for Fibonacci, Lucas polynomials and more.

\section{Some properties of the generalized Bernoulli polynomials and lemmas}\label{sec:some}

Let us consider the three following operators defined over any endomorphism of the vector space $\mathbb{C}\left[ x\right]$.
The classical derivation operator $D$, the identity operator $I$, and the finite difference operator $\Delta$ are respectively defined by
\begin{equation}
I(x^{n})=x^{n}\text{ and } \Delta (x^{n})=(x+1)^{n}-x^{n},\text{\ }n\in \mathbb{N}.
\end{equation}
Generalized Bernoulli polynomials can be expressed as a sequence of Appell polynomials \cite{appl,ben2}. These polynomials satisfy the following well-known properties for which the proofs are straightforward \cite{rom}
\begin{align}
B_{0}^{(\alpha)}(x)&amp;=1,\\
D(B_{n}^{(\alpha)}(x))&amp;=nB_{n-1}^{(\alpha)}(x),\text{\ }n\geq 1,  \label{f14}\\
B_{n}^{(\alpha)}(x+y)&amp;=\sum_{k=0}^{n}\binom{n}{k}y^{n-k}B_{k}^{(\alpha)}(x), \label{f15}\\
\Delta (B_{n}^{(\alpha)}(x))&amp;=D(B_{n}^{(\alpha-1)}(x)), \label{f13}\\
B_{n}^{(\alpha)}(\alpha-x)&amp;=(-1)^{n}B_{n}^{(\alpha)}(x).  \label{f30}
\end{align}
For every $\alpha \in \mathbb{C}$, let us consider the endomorphism $\Omega_{\alpha }$ of $\mathbb{C} \left[x\right]$ defined by
\begin{equation}
\Omega_{\alpha}(x^{n})=B_{n}^{(\alpha)}(x),\text{\ }n\in \mathbb{N}.
\end{equation}
\begin{lemma}
		\label{lem4}For every non-negative integer $n$ and for all complex numbers $\alpha$ and $\gamma$, we have
		\begin{equation}
		\Omega_{\alpha}((x+\gamma)^{n})=B_{n}^{(\alpha)}(x+\gamma). \label{f12}
		\end{equation}
\end{lemma}
\begin{proof}
		Just by using Property (\ref{f15}) for $y=\gamma$, it follows that
		\begin{equation}
		\Omega_{\alpha}(( x+\gamma)^{n})=\Omega_{\alpha
		}\bigg(\sum_{k=0}^{n}\binom{n}{k}\gamma^{n-k}x^{k}\bigg)=\sum_{k=0}^{n}
		\binom{n}{k}\gamma^{n-k}B_{k}^{(\alpha)}(x)
		=B_{n}^{(\alpha)}(x+\gamma).
		\end{equation}
\end{proof}
The operators $\Omega_{\alpha}$, $\Delta$, and $D$ satisfy some remarkable
relations that are mentioned in the following lemma.
\begin{lemma}
		\label{lem1}For every $\alpha \in \mathbb{C}$, we have
		\begin{align}
		D\circ \Omega_{\alpha}&amp;=\Omega_{\alpha}\circ D, \label{t9}\\
		\Omega_{\alpha}\circ \Delta&amp;=\Omega_{\alpha-1}\circ D. \label{t6}
		\end{align}
\end{lemma}
\begin{proof}
		\begin{enumerate}
			\item It is sufficient to verify that the equality $(D\circ \Omega_{\alpha
			})(x^{n})=(\Omega_{\alpha}\circ D)(
			x^{n})$ is true for all non-negative integers $n$. For $n=0$, the property is obvious since
            $B_{0}^{(\alpha)}(x)=1$ and thus $(D\circ \Omega_{\alpha})(x^{0})=0=(
			\Omega_{\alpha}\circ D)(x^{0})$. For every $n\geq 1$, we have
			\begin{equation}
			(D\circ \Omega_{\alpha})(x^{n})=D(
			B_{n}^{(\alpha)}(x))=\Omega_{\alpha}(nx^{n-1})=(\Omega_{\alpha}\circ D)(x^{n}).
			\end{equation}
			\item By using Lemma \ref{lem4}, for $\gamma =1$, Equations (\ref{f13}) and (\ref{t9}), for all integers
              $n\geq 0$, it follows that
	         \begin{align*}
				(\Omega_{\alpha}\circ \Delta)(x^{n})&amp;=\Omega_{\alpha}((x+1)^{n}-x^{n}) \\
				&amp;=B_{n}^{(\alpha)}(x+1)-B_{n}^{(\alpha)}(x) \\
				&amp;=\Delta(B_{n}^{(\alpha)}(x)) \\
				&amp;=D(B_{n}^{(\alpha-1)}(x)) \\
				&amp;=(D\circ \Omega_{\alpha -1})(x^{n}) \\
				&amp;=(\Omega_{\alpha -1}\circ D)(x^{n}).
			\end{align*}
		\end{enumerate}
\end{proof}
Abramowitz and Stegun \cite{abr} presented the well-known following properties of classical Bernoulli numbers and Bernoulli polynomials:
\begin{align}
B_{n}(x+1)&amp;=\sum_{k=0}^{n}\binom{n}{k}B_{k}(x)\notag \\
&amp;=B_{n}(x)+nx^{n-1},\text{\ }n\geq 1,  \label{ff1}\\
\sum_{k=0}^{n}\binom{n}{k}B_{k}&amp;=B_{n}+\delta_{n,1}, \label{f20}\\
(-1)^{n}B_{n}&amp;=B_{n}+\delta_{n,1},  \label{f21}
\end{align}
where $\delta_{i,j}$ is Kronecker's symbol taking the value $1$ if $i=j$ and $0$
otherwise. Notice that Eq.~(\ref{f21}) is equivalent to
\begin{equation}
B_{1}=-\frac{1}{2}\quad \text{and}\quad B_{2n+1}=0,\text{\ }n\geq 1. \label{f22}
\end{equation}
By using the relations above, we can calculate the first Bernoulli numbers and Bernoulli polynomials.

\section{Literature review}\label{sec:lit}

One notices that Relations (\ref{t4}) and (\ref{t5}) proven by Gessel, can be reformulated in a more useful form for this study. Changing $k$ to $m-k$ in the right side of Relation (\ref{t4}), one obtains the following identity:
\begin{equation}
S_{n,l,r}^{(1)}(m,0,0)=(r+1)\sum_{k=1}^{m-1}\sum_{j=0}^{r+1}\binom{n+r}{j}\binom{\ell+r}{r+1-j}
k^{n+r-j}(k-m)^{\ell +j-1}. \label{tg4}
\end{equation}
So, noting that for all integers $r$ and $s$ such that $r+s$ is even, we deduce the following property.
\begin{equation}
\sum_{k=1}^{m-1}(k^{r}(k-m)^{s}-k^{s}(k-m)^{r})=0,  \label{rem1}
\end{equation}
A\"{\i}der and Bencherif \cite{aid1} proposed a formula
equivalent to Gessel's identity, which
can be written as follows: \begin{equation}
\sum_{k=0}^{n+r}m^{n+r-k}\binom{n+r}{k}\binom{n+k+r}{r}B_{n+k}=\sum_{k=1}^{m-1}p_{r}(n,m,k),
\label{ges1} \end{equation} where \begin{align}
p_{r}(n,m,k)&amp;=\frac{r+1}{2}\binom{n+r}{\frac{r+1}{2}}^{2}(k(k-m))^{n+\frac{r-1}{2}}
\notag \\ &amp;+(r+1)\sum_{j=0}^{\frac{r-1
}{2}}\binom{n+r}{j}\binom{n+r}{r+1-j}k^{n+r-j}(k-m)^{n+j-1}.
\label{tg5} \end{align} 
In what follows, we show that Gessel's relations
(\ref{t4}) and (\ref{t5}) and their equivalent relations (\ref{tg4}) and
(\ref{ges1}) generalize numerous identities involving Bernoulli numbers.
Therefore, from Equations (\ref{f20}) and (\ref{f21}), Bencherif
and Garici \cite{ben1} stated the following well-known property:
\begin{equation} \sum_{k=0}^{n}\binom{n}{k}B_{k}=(-1)^{n}B_{n},
\label{p1} \end{equation} which can be written as \begin{equation}
S_{n,0,0}^{(1)}(1,0,0)=0.  \end{equation} Eq.~(\ref{p1}) means that
the sequence $((-1)^{n}B_{n})_{n\geq 0}$ is self-dual (in other
words, a Cesaro sequence) \cite{ben1,chen,chu,mu,sun1,sun,zek2}
and \cite[p.~256]{luc2}. Gould \cite{gou1} also presented various
explicit formulae for Bernoulli numbers.  However, several authors have
sought other recurrence relations that would make simpler Bernoulli
numbers computation.  Thus, in 1880 Lucas \cite{luc1} proved, by
using symbolic calculation, the following relation that he deemed
important: \begin{equation} \sum_{k=0}^{n}\binom{n}{k}B_{\ell
+k}+(-1)^{\ell +n+1}\sum_{k=0}^{\ell}\binom{\ell}{k}B_{n+k}=0,
\label{e1} \end{equation} which can be written as \begin{equation}
S_{n,\ell,0}^{(1)}(1,0,0)=0. \label{e2} \end{equation} This relation
therefore constitutes a special case of Eq.~(\ref{t3}). Lucas then
noticed that for the special case where $\ell=n+1$, the Eq.~(\ref{e2})
becomes \begin{equation} (n+1)S_{n,n+1,0}^{(1)}(1,0,0)=\sum_{k=0}^{n+1}
\binom{n+1}{k}(n+k+1)B_{n+k}=0,  \label{k5} \end{equation}
furthermore, we deduce that for $n\geq 1$ \begin{equation}
\sum_{k=0}^{n}\binom{n+1}{k}(n+k+1)B_{n+k}=0.  \label{k3} \end{equation}
According to Lucas, Eq.~(\ref{k3}) is a great use for the computation
Bernoulli numbers.  It permits the expression of the Bernoulli
number $B_{2n}$ just by using Bernoulli numbers $B_{j}$ for $n\leq
j\leq 2n-1$, which requires less computation than by applying the
following relation deduced from Eq.~(\ref{f20}) \begin{equation*}
B_{2n}=\frac{-1}{2n+1}\sum_{j=0}^{2n-1}\binom{2n+1}{j}B_{j}.
\end{equation*} This latter assumes the knowledge of Bernoulli
numbers $B_{j}$ for $0\leq j\leq 2n-1$. In fact, Property (\ref{k3})
was also discovered in 1827 by Von Ettingshausen \cite{ett} before
being rediscovered in 1877 by Seidel \cite{sei}.  Lucas proved Relation
(\ref{e1}) using symbolic calculation or umbral calculus as shown in the
articles of Agoh \cite{ago1} and Gessel \cite{ges}.  This relation has
been the subject of intensive research. After Lucas, several authors have
proved Eq.~(\ref{k3}) again, using different methods. Then, Nielsen
\cite{nie} proved the same relation again in 1923, as Kaneko \cite{kan}
did in 1995. Kaneko provided two different proofs of Relation (\ref{k3}),
one complicated one based on the theory of continued fractions applied to
formal series, and the second much simpler one due to Zagier, based on an
involutive transform of sequences. The relation was later proven again in
2000 by Satoh \cite{sat}, then in 2001 by Chang and Ha \cite[Corollary.\ 1
(a), p.\ 472]{cha}, and afterward in 2009 by Cigler \cite{cig}.

Recall that
in 1971, Carlitz \cite{car} already wondered whether Relation (\ref{e1})
could be deduced from only Eq.~(\ref{f20}). A year later, in response
to Carlitz's problem, Shannon \cite{sha} proved by using 
Eq.~(\ref{f22}) by induction on $m$ and $n$.
In 2005, Vassilev and Missana \cite{vas2} demonstrated the
following relation: \begin{equation} \sum_{k=0}^{n-1}\binom{n}{k}B_{\ell
+k}+(-1)^{\ell+n+1}\sum_{k=0}^{\ell -1}\binom{\ell}{k}B_{n+k}=0,\text{
for }n\geq 1\ \text{\ and }\ell \geq 1, \end{equation} which can be written
as \begin{equation} S_{n,\ell,0}^{(1)}(1,0,0)-(1-(-1)^{\ell+n})B_{n+\ell
}=0,\text{ for }n\geq 1\  \text{and } \ell \geq 1, \end{equation} which
is equivalent to Eq.~(\ref{e1}). Their proof refers to the symmetry of
polynomials of two variables involving Bernoulli numbers, introduced
in \cite{vas1}.

In 2007, Chu and Magli \cite{chu} again proved Eq.~(\ref{e1}) by
exploiting Eq.~(\ref{p1}). The same identity had been shown in 2009 by
Chen and Sun \cite{che} via a computational algebraic approach, using
an extension of Zeilberger's algorithm. In 2012, Bencherif and Garici
\cite{ben1} proved Eq.~(\ref{e1}) by using Eq.~(\ref{p1}). In 2014, Gould
and Quaintance \cite{gou}, by studying some properties of the binomial
transform of a sequence, obtained the same result. In the same year,
Prodinger \cite{pro} presented a short proof based on a special case of
generating functions. A year later, Neto \cite{net} managed to get the
same result by exploiting some properties of the Zeon algebra. In the
same year, Zekiri and Bencherif \cite{zek1} attained a more general result,
by applying the umbral calculus.

Recall that in 2000, Agoh \cite[Eq.\ (4.3), p.\ 210]{ago1} showed the generalization of Eq.~(\ref{e2}) 
\begin{equation}
S_{n,\ell,r}^{(1)}(1,0,0)=0  \label{s3}
\end{equation}
by using congruences and umbral calculus.
In the following year, Momiyama \cite{mom} proved a formula
equivalent to Eq.~(\ref{s3}) for $r=1$, by applying a $q$-adic method. Shortly after, in 2003, in his article on
the applications of classical umbral calculus, Gessel \cite{ges}
produced the generalized Eq.~(\ref{k5}), which he called Kaneko's identity.
First, Gessel began by examining the relation below, which is none other than
Formula (\ref{tg4}) when $r=0$:
\begin{equation}
S_{n,\ell,0}^{(1)}(m,0,0)=\sum_{k=1}^{m-1}((n+\ell )k-mn)k^{n-1}(k-m)^{\ell-1}.
\end{equation}
Then, by using the following property,
\begin{equation}
(n+1)S_{n,n+1,0}^{(1)}(m,0,0)=\sum_{k=0}^{n+1}m^{n+1-k}\binom{n+1}{k}(n+k+1)B_{n+k},  \label{t24}
\end{equation}
he obtained the following generalization of Kaneko's identity:
\begin{equation}
\sum_{k=0}^{n+1}m^{n+1-k}\binom{n+1}{k}(n+1+k)B_{n+k}=\sum_{k=1}^{m-1}p_{1}(m,n,k),
\end{equation}
which is exactly Formula (\ref{ges1}) when $r=1$.

In 2009, Chen and Sun \cite{che} proved, by using an extension of
Zeilberger's algorithm, that
\begin{equation}
\sum_{k=0}^{n+3}m^{n+3-k}\binom{n+3}{k}\binom{n+k+3}{3}B_{n+k}=\sum_{k=1}^{m-1}q(m,n,k). \label{c1}
\end{equation}
Then, Chen and Sun provided the expression $q(m,n,k)$, which can be written as follows:
\begin{equation*}
q(m,n,k)=p_{3}(m,n,k)+\binom{n+3}{3}(3n+11)(k^{n+2}(k-m)^{n}-k^{n}(k-m)^{n+2}).
\end{equation*}
Recalling Eq.~(\ref{rem1}), one notices that
\begin{equation}
\sum_{k=1}^{m-1}q(m,n,k)=\sum_{k=1}^{m-1}p_{3}(m,n,k).
\end{equation}
Equality (\ref{c1}) is equivalent to Formula (\ref{ges1}) for $r=3$.

\section{Main result}\label{sec:main}

The following theorem provides some simplified expressions of $S_{n,\ell
,r}^{(\alpha)}(x,y,z)$ for $x+y+z-\alpha=s$ where $s$ is a non-negative
integer. Then it leads to interesting identities for $s=0$ or $\alpha=1$.
\begin{theorem}\label{theo}
For all complex numbers $\alpha$, $\lambda$, and for all non-negative
integers $l$, $n$, $r$, and $s$, we have
     \begin{align}
         &amp;\sum_{k=0}^{n+r}\lambda^{n+r-k}\binom{n+r}{k}\binom{\ell +k+r}{r}B_{\ell
            +k}^{(\alpha)}(x)  \notag \\
         &amp;+(-1)^{\ell+n+r+1}\sum_{k=0}^{\ell +r}\lambda^{\ell +r-k}\binom{\ell+r}
            {k}\binom{n+k+r}{r}B_{n+k}^{(\alpha)}(\alpha+s-\lambda-x)  \notag \\
         &amp;=\Omega_{\alpha-1}(\frac{D^{r+1}}{r!}\sum_{k=1}^{s}(x-k)^{\ell+r}(x+\lambda-k)^{n+r}).  \label{le1}
     \end{align}
\end{theorem}

\begin{proof}
Let us consider the polynomial $P(x)$ defined by
\begin{equation*}
P(x)=\sum_{k=1}^{s}P_{k}(x)
\end{equation*}
where
\begin{equation*}
P_{k}(x)=\frac{D^{r}}{r!}\bigr((x-k)^{\ell+r}(x+\lambda-k)^{n+r}\bigr).
\end{equation*}
One notices that the right side of Eq.~(\ref{le1}) is
equivalent to $(\Omega_{\alpha-1}\circ D)(P(x))$, and from Lemma \ref{lem1}, one has
$(\Omega_{\alpha-1}\circ D)(P(x))
=(\Omega_{\alpha }\circ \Delta)(P(x))$.
To prove Theorem \ref{theo}, it suffices to show that
$(\Omega_{\alpha}\circ \Delta)(P(x))$
equals to the left side of Eq.~(\ref{le1}). For this, one notes the equality
$P_{k}(x+1)=P_{k-1}(x)$ and thus
\begin{equation}
(\Omega_{\alpha}\circ \Delta)(P(x))=\Omega_{\alpha}\bigg(\sum_{k=1}^{s}(P_{k-1}(x)
-P_{k}(x))\bigg)=\Omega_{\alpha}(P_{0}(x))-\Omega_{\alpha}(P_{s}(x)).  \label{abc1}
\end{equation}
One has
\begin{align*}
P_{0}(x)&amp;=\frac{D^{r}}{r!}x^{\ell+r}(x+\lambda)^{n+r} \\
&amp;=\frac{D^{r}}{r!}\sum_{k=0}^{n+r}\lambda^{n+r-k}\binom{n+r}{k}x^{\ell+k+r} \\
&amp;=\sum_{k=0}^{n+r}\lambda ^{n+r-k}\binom{n+r}{k}\binom{\ell+k+r}{r}x^{\ell+k}
\end{align*}
and
\begin{align*}
P_{s}(x)&amp;=\frac{D^{r}}{r!}((x+\lambda -s)-\lambda)^{\ell +r}(x+\lambda -s)^{n+r} \\
&amp;=\frac{D^{r}}{r!}\sum_{k=0}^{n+r}(-\lambda)^{\ell+r-k}\binom{\ell+r}{k}( x+\lambda -s)^{n+k+r} \\
&amp;=(-1)^{\ell +n+r}\sum_{k=0}^{n+r}\lambda ^{\ell+r-k}\binom{\ell+r}{k}\binom{n+k+r}{r}(-1)^{n+k}(x+\lambda -s)^{n+k}.
\end{align*}
Equations (\ref{f30}) and (\ref{f12}) yield 
\begin{equation}
\Omega_{\alpha}(P_{0}(x))=\sum_{k=0}^{n+r}\lambda^{n+r-k}{n+r}{k}\binom{\ell+k+r}{r}B_{\ell+k}^{(\alpha)}(x)  \label{abc2}
\end{equation}
and
\begin{align}
\Omega_{\alpha}(P_{s}(x))&amp;=(-1)^{\ell+n+r}\sum_{k=0}^{n+r}\lambda^{\ell+r-k}\binom{\ell +r}{k}\binom{
n+k+r}{r}(-1)^{n+k}B_{n+k}^{(\alpha)}(x+\lambda -s)  \notag \\
&amp;=(-1)^{\ell+n+r}\sum_{k=0}^{\ell+r}\lambda^{\ell +r-k}\binom{\ell+r}{k}
\binom{n+k+r}{r}B_{n+k}^{(\alpha)}(\alpha +s-\lambda -x).\label{abc3}
\end{align}
By using Equations (\ref{abc1}), (\ref{abc2}), and (\ref{abc3}), it
follows that $( \Omega _{\alpha }\circ \Delta)(P(x))$ is indeed equal to
the left side of Eq.~(\ref{le1}). This completes our proof.  \end{proof}

\section{Applications}\label{sec:App}

As already seen in the previous section, Theorem \ref{theo} is an
extension of Gessel's formula, since it generalizes several
identities involving Bernoulli numbers. In this section, we illustrate how
Theorem \ref{theo} allows us to obtain more identities involving Bernoulli
polynomials. For this purpose, noticing that by using Eq.~(\ref{f30}), one has
\begin{equation*}
B_{n+k}^{(\alpha)}(x) =(-1)^{n+k}B_{n+k}^{(\alpha)}(\alpha -x)
\end{equation*}
and thus $S_{n,l,r}^{(\alpha)}(x,y,z)$ given by Eq.~(\ref{t1}) becomes as follows:
\begin{align*}
S_{n,l,r}^{(\alpha)}(x,y,z)&amp;=\sum_{k=0}^{n+r}\binom{n+r}{k}\binom{\ell+k+r}{r}x^{n+r-k}B_{\ell +k}^{(\alpha)}(y) \\
&amp;(-1)^{\ell+r+1}\sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}\binom{n+k+r}{r}x^{\ell+r-k}B_{n+k}^{(\alpha)}(\alpha -z).
\end{align*}
Also note that Eq.~(\ref{le1}) can be formulated as follows:
\begin{equation*}
S_{n,l,r}^{(\alpha)}(\lambda ,x,\alpha +s-\lambda-x) =\Omega_{\alpha -1}
\bigg(\frac{D^{r+1}}{r!}\sum_{k=1}^{s}(x-k)^{\ell +r}(x+\lambda -k)^{n+r}\bigg).
\end{equation*}
The left side of Eq.~(\ref{le1}) can be written as
\begin{align*}
&amp;\sum_{k=0}^{n+r}\lambda ^{n+r-k}\binom{n+r}{k}\binom{\ell+k+r}{r}B_{\ell+k}^{(\alpha)}(x) \\
&amp;-\sum_{k=0}^{\ell +r}(-1)^{\ell +r+k}\lambda ^{\ell+r-k}\binom{\ell+r}{k}\binom{n+k+r}{r}
B_{n+k}^{(\alpha)}(x+\lambda -s).
\end{align*}
By applying Leibniz's formula for $\alpha =1$, the left
side of Eq.~(\ref{le1}) can be written as follows:
\begin{equation*}
(r+1)\sum_{k=1}^{s}\sum_{j=0}^{r+1}\binom{n+r}{j}\binom{\ell+r}{r+1-j}
(x-k)^{l+j-1}(x+\lambda -k) ^{n+r-j}
\end{equation*}
and equals zero if $s=0$.

\subsection{Generalization of Gessel's formula to classical
Bernoulli polynomials}

For $\alpha =1$, we have $B_{n}^{(1)}(x)
=B_{n}(x)$ and $\Omega _{\alpha -1}=\Omega _{0}=I$. Recalling
Theorem \ref{theo}, for $\alpha =1$, $\lambda =m$, and $s=m-1$ where $m$
is a non-negative integer, leads us to the following corollary, which is a
generalization of Gessel's theorem set in \cite{bel}.
\begin{corollary}
	\label{coro1} For every non-negative integers $\ell$, $n$, $r$, and $m$, we have
	\begin{align}
		&amp;\sum_{k=0}^{n+r} m^{n+r-k}\binom{n+r}{k}\binom{\ell+k+r}{r}B_{\ell+k}(x) \notag \\
		&amp;+(-1)^{\ell+n+r+1}\sum_{k=0}^{\ell +r} m^{\ell+r-k}\binom{\ell+r}{k}\binom{n+k+r}{r}B_{n+k}(-x) \notag \\
		&amp;=(r+1)\sum_{k=1}^{m-1}\sum_{j=0}^{r+1}\binom{n+r}{j}
          \binom{\ell+r}{r+1-j}(x-k)^{\ell +j-1}(x+m-k)^{n+r-j}.  \label{cor}
	\end{align}
\end{corollary}
Replacing $x$ by $0$ in Eq.~(\ref{cor}), we actually find Formula (\ref{t4}) thanks to Gessel. One notes that He \cite{he2}
found, in 2014, the Gessel formula by using $q$-numbers and Bernoulli polynomials.

\subsection{Nielsen's identity (1923)}

First, let us consider the formula proposed by Nielsen \cite{nie} and
published in 1923, where Bernoulli numbers and polynomials are differently defined but they match
our notation $(-1)^{n-1}B_{2n}$ and $\frac{B_{n}(x+1)}{n!}$ respectively.
Then, the equation in \cite[Eq.\ (10), p.\ 182]{nie} can, with an adjusted notation, be stated as follows:
\begin{align}
&amp;\sum_{k=0}^{n+r}(1-2\beta)^{n+r-k}\binom{n+r}{k}\binom{\ell+k+r}{r}B_{\ell+k}(x+\beta)  \notag \\
&amp;-\sum_{k=0}^{\ell +r}(-1)^{\ell +r+k}(1-2\beta)^{\ell +r-k}\binom{\ell+r}{k}\binom{n+k+r}{r}B_{n+k}(x-\beta)  \notag \\
&amp;=\frac{D^{r+1}}{r!}\bigg((x+\beta -1)^{\ell+r}(x-\beta)^{n+r}\bigg). \label{edf1}
\end{align}
By using Eq.~(\ref{f30}), Formula (\ref{edf1}) can be written as
\begin{equation*}
S_{n,\ell,r}^{(1)}(1-2\beta,x+\beta,1-x+\beta)=\frac{D^{r+1}}{r!}\bigg(( x+\beta -1)^{\ell +r}(x-\beta)^{n+r}\bigg),
\end{equation*}
which is true according to Theorem \ref{theo}.

\subsection{Agoh's identities (2000)}

In 2000, Agoh \cite{ago1} investigated linear recurrences for Bernoulli numbers and Bernoulli polynomials. He derived
many identities including the two following equations, \cite[Eq.\ (3.2)(i), p.\ 205]{ago1} and \cite[Eq.\ (3.4)(i), p.\ 207]{ago1}:
\begin{align}
&amp;\sum_{k=0}^{n}m^{n-k}\binom{n}{k}B_{\ell +k}(x)
-\sum_{k=0}^{\ell }(-m)^{\ell-k}\binom{\ell}{k}B_{n+k}(x)  \notag \\
&amp;=n\sum_{k=0}^{m-1}(x+k)^{n-1}(x-m+k)^{\ell}+
\ell\sum_{k=0}^{m-1}(x+k)^{n}(x-m+k)^{\ell-1}  \label{reda1}
\end{align}
and
\begin{align}
&amp;\sum_{k=0}^{n+r}\binom{n+r}{k}\binom{\ell+k+r}{r}B_{\ell +k}(x)  \notag \\
&amp;+(-1)^{\ell+r+1}\sum_{k=0}^{\ell +r}(-1)^{k}\binom{\ell+r}{k}\binom{n+k+r}{r}B_{n+k}(x)  \notag \\
&amp;=(n+r)\sum_{k=0}^{r}\binom{n+r-1}{k}\binom{\ell+r}{r-k}
x^{n+r-k-1}(x-1)^{\ell +k}  \notag \\
&amp;+(\ell+r)\sum_{k=0}^{r}\binom{\ell+r-1}{k}\binom{n+r}{r-k}
x^{n+k}(x-1)^{\ell +r-k-1}. \label{edf2}
\end{align}
Theorem \ref{theo} allows us to directly obtain these two identities. Indeed,
according to this theorem, for all non-negative integers $m$, we have
\begin{equation}
S_{n,\ell,0}^{(1)}(m,x,-x)=D\bigg(\sum_{k=1}^{m}(x-k)^{\ell}(x+m-k)^{n}\bigg).
\label{yese}
\end{equation}
Changing $k$ to $m-k$ in Eq.~(\ref{yese}), one has
\begin{align*}
D\bigg(\sum_{k=1}^{m}(x-k)^{\ell}(x+m-k)^{n}\bigg)&amp;=D\bigg(\sum_{k=0}^{m-1}(x+k)^{n}(x-m+k)^{\ell}\bigg)\\
&amp;=n\sum_{k=0}^{m-1}(x+k)^{n-1}(x-m+k)^{\ell}\\
&amp;+\ell\sum_{k=0}^{m-1}(x+k)^{n}(x-m+k)^{\ell-1},
\end{align*}
and this is just Eq.~(\ref{reda1}).
For $\beta =0$, and by noticing that the right side of Eq.~(\ref{edf1}) can
be rewritten as
\begin{align*}
&amp;\frac{D^{r}}{r!}\bigg(( n+r)x^{n+r-1}(x-1)^{\ell+r}+(\ell+r)x^{n+r}(x-1)^{\ell+r-1}\bigg) \\
&amp;=(n+r)\sum_{k=0}^{r}\binom{n+r-1}{k}\binom{\ell+r}{r-k}x^{n+r-k-1}(x-1)^{\ell +k} \\
&amp;+(\ell+r)\sum_{k=0}^{r}\binom{\ell+r-1}{k}\binom{n+r}{r-k}x^{n+k}(x-1) ^{\ell+r-k-1},
\end{align*}
one obtains, with an adjusted notation, Eq.~(\ref{edf2}).
It should be noted that for $r\in  \{0,1 \}$, Eq.~(\ref{edf2}) becomes the following
formulae \cite[Cor.\ 3.4, Eq.\ (3.9), Eq.\ (3.10), p.\ 163]{ago2} which is what Agoh found in 2017, through a different method:
\begin{equation*}
\sum_{k=0}^{n}\binom{n}{k}B_{\ell +k}(x)-\sum_{k=0}^{\ell}(-1)^{\ell-k}\binom{\ell}{k}B_{n+k}(x)=
\bigr((n+\ell)x-n\bigr)x^{n-1}\bigr( x-1\bigr)^{\ell -1}
\end{equation*}
and
\begin{align*}
&amp;\sum_{k=0}^{n+1}\binom{n+1}{k}(\ell+k+1)B_{\ell +k}(x)-\sum_{k=0}^{\ell +1}(-1)^{\ell +1-k}\binom{\ell+1}{k}
(n+k+1)B_{n+k}(x) \\
&amp;=\bigr((n+\ell +2)(n+\ell +1)x^{2}-2(n+1)(n+\ell+1)x+(n+1)n\bigr)x^{n-1}\bigr(x-1\bigr)^{\ell-1}.
\end{align*}

\subsection{Chang and Ha's identity (2001)}

In 2001, Chang and Ha \cite{cha} obtained a class of recurrence relations
for the Bernoulli numbers that includes the Kaneko formula (\ref{k3}).
Among these identities, we can find the following relations due to Chang and Ha \cite[Cor.\ 1.b, p.\ 472]{cha}:
\begin{equation}
\sum_{k=n}^{2n}\binom{n+1}{k-n}(k+1)\frac{B_{k}}{2^{k}}=(-1)^{n}\frac{n+1}{2^{2n+1}}, n\geq 1.  \label{chaaa2}
\end{equation}
Below, we show that Theorem \ref{theo} allows us to retrieve Eq.~(\ref{chaaa2}). From Theorem \ref{theo}
\begin{equation*}
S_{n,n,r}^{(1)}\bigl(2-2\beta,\beta+x, \beta-x\bigl)
=\bigl(r+1\bigl)\frac{D^{r+1}}{(r+1)!}\bigl( x^{2}-(\beta -1)^{2}\bigl)^{n+r}.
\end{equation*}
For $x=0$ and $r$ odd, one deduces that
\begin{align*}
2\sum_{k=0}^{n+r}(2-2\beta)^{n+r-k}\binom{n+r}{k}&amp;\binom{n+k+r}{r}B_{n+k}(\beta) \\
&amp;=(-1)^{n+\frac{r-1}{2}}(r+1)\binom{n+r}{\frac{r+1}{2}}(\beta-1)^{2n+r-1}.
\end{align*}
Then for $\beta =0$, we have
\begin{equation}
\sum_{k=0}^{n+r}\binom{n+r}{k}\binom{n+k+r}{r}\frac{B_{n+k}}{2^{n+k}}=\frac{
(-1)^{n+\frac{r-1}{2}}(r+1)}{2^{2n+r+1}}\binom{n+r}{\frac{r+1}{2}}.  \label{chaaa1}
\end{equation}
For $r=1$, changing $k$ to $k-n$ in Eq.~(\ref{chaaa1}), we obtain Eq.~(\ref{chaaa2}).

\subsection{Sun's identities (2003)}

In 2003, Sun \cite{sun} derived a general combinatorial identity in
terms of polynomials with dual sequences of coefficients as he deduced
combinatorial identities involving Bernoulli polynomials, including
following relations: \cite[Thm.\ 1.2, Eq.\ (1.15), Eq.\ (1.16), p.\ 712]{sun} and \cite[Remark.\ 1.2, Eq.\ (1.18) p.\ 713]{sun},
where $z=1-x-y$.
\begin{equation}
(-1)^{n}\sum_{k=0}^{n}\binom{n}{k}x^{n-k}B_{\ell +k}(y)
=(-1)^{\ell}\sum_{k=0}^{\ell}\binom{\ell}{k}x^{\ell -k}B_{n+k}(z)=0,  \label{cha3}
\end{equation}
\begin{align}
&amp;(-1)^{n}\sum_{k=0}^{n}\binom{n+1}{k}(\ell+k+1)
x^{n+1-k}B_{\ell +k}(y)  \notag \\
&amp;+(-1)^{\ell}\sum_{k=0}^{\ell}\binom{\ell+1}{k}(
n+k+1)x^{\ell+1-k}B_{n+k}(z)  \notag \\
&amp;=(-1)^{n}(n+\ell+2)\bigl((B_{n+\ell +1}(
x+y)) -B_{n+\ell +1}(y)\bigl),  \label{reda4}
\end{align}
\begin{equation}
\sum_{k=0}^{n}\binom{n+1}{k}(n+k+1)(1-2x)^{n-k+1}B_{n+k}(x)=-2(n+1)B_{2n+1}(x).  \label{chel6}
\end{equation}
Theorem \ref{theo} allows us to deduce the three identities above. Indeed,
the theorem states that $S_{n,\ell,r}^{(1)}(x,y,z) =0$ for $x+y+z=1$. And for $r=0$, it gives Eq.~(\ref{cha3}).
Chen and Sun \cite[Thm.\ 5.1, p.\ 2121]{che} proved this equation. If $r=1$, it becomes a relation
from which Eq.~(\ref{reda4}) can be derived. One notices that
\begin{equation*}
B_{n+k}(z)=B_{n+k}(1-x-y)=(-1)^{n+k}B_{n+k}(x+y).
\end{equation*}
Finally, by Theorem \ref{theo}, we also have $\frac{1}{2}
S_{n,n,1}^{(1)}(1-2x,x,x)=0$, i.e., Eq.~(\ref{chel6}).
Note that, in 2016, Pita \cite{pit} also proved Eq.~(\ref{cha3}).

\subsection{Wu, Sun, and Pan's identities (2004)}

In 2004, by studying some formal power series, Wu, Sun, and Pan \cite{wu}
obtained the following identities \cite[Thm.\ 2, Eq.\ (6), Eq.\ (8), p.\ 3]{wu}:
\begin{equation}
\text{ }(-1)^{n}\sum_{k=0}^{n}\binom{n}{k}B_{\ell+k}(x)
=(-1)^{\ell}\sum_{k=0}^{\ell}\binom{\ell}{k}B_{n+k}(-x),  \label{cha5}
\end{equation}
and
\begin{align}
&amp;\left(-1\right)^{n}\sum_{k=0}^{n}\binom{n+1}{k}(\ell+k+1)
B_{\ell +k}(x)+(-1)^{\ell}\sum_{k=0}^{\ell}
\binom{\ell+1}{k}(n+k+1)B_{n+k}(-x)  \notag \\
&amp;=(-1)^{n}(n+\ell+2)(n+\ell+1)
x^{n+\ell}.  \label{cha11}
\end{align}

These identities can be obtained promptly by using Theorem \ref{theo}, especially if we have
$S_{n,\ell ,r}^{(1) }(1,x,-x) =0$.
Replacing $r$ by $0$ in Theorem \ref{theo}, we obtain Eq.~(\ref{cha5}); however, if $r=1$, Equations (\ref{f30}), (\ref{ff1}),
and Theorem \ref{theo} lead to Eq.~(\ref{cha11}).

\subsection{Chen's identity (2007)}

A sequence $(b_{n})_{n\geq 0}$ is called a ``binomial transform'' of another sequence $(a_{n})_{n\geq 0}$ if and only if
$b_{n}=\sum_{k=0}^{n}\binom{n}{k}a_{k}$.

In 2007, Chen \cite{chen}
proposed a general identity for such pairs of sequences from which he deduced several
identities for Bernoulli numbers and polynomials. One of these identities is the
relation \cite[Thm.\ 5.3, Eq.\ (5.7), p.\ 149]{chen}:
\begin{align}
&amp;\sum_{k=0}^{n+r}\binom{n+r}{k}\binom{\ell+k+r}{r}y^{n+r-k}B_{\ell+k}(x)  \notag \\
&amp;=\sum_{k=0}^{\ell+r}\binom{\ell+r}{k}\binom{n+k+r}{r}(-y)^{\ell+r-k}B_{n+k}(x+y), \label{cha10}
\end{align}
which can be written as $S_{n,\ell,r}^{(1)}(y,x,1-x-y)=0$.
This can be obtained directly from Theorem \ref{theo}. Note that, in 2010, He and Zhang \cite{he1} also proved Eq.~(\ref{cha10}).

\subsection{Neto's identity (2015)}

In 2015, Neto \cite{net} provided a short proof of Eq.~(\ref{e1}) using 
Zeon algebra, and then proved
the analogous identity for the Bernoulli numbers of higher order:
\begin{equation}
\sum_{k=0}^{n}x^{n-k}\binom{n}{k}B_{\ell+k}^{(x)}=(-1)^{n+\ell}\sum_{k=0}^{\ell }x^{\ell-k}
\binom{\ell}{k}B_{n+k}^{(x)}. \label{re1}
\end{equation}
This identity can be written as  $S_{n,\ell,0}^{(x)}(x,0,0)=0$, which can be obtained immediately from Theorem \ref{theo}. In
fact, it is interesting to express Relation (\ref{re1}) by using Stirling polynomials. Indeed, Stirling polynomials $\sigma_{n}(x)$ are defined in \cite{gra} by
\begin{equation*}
\bigg(\frac{te^{t}}{e^{t}-1}\bigg)^{x}=x\sum_{n=0}^{\infty}\sigma
_{n}(x)t^{n}.
\end{equation*}
One has
\begin{equation}
n!x\sigma _{n}(x)=(-1)^{n}B_{n}^{(x)}. \label{cha8}
\end{equation}
$B_{n}^{(x)}$ are the generalized Bernoulli numbers, which are also called 
N\"orlund polynomials \cite{ben0}.

More generally, from Theorem \ref{theo}, we have
\begin{equation}
S_{n,\ell,r}^{(x)}(x,0,0)=0,  \label{cha9}
\end{equation}
for all non-negative integers $r$. By Equations (\ref{cha8}) and (\ref{cha9}), one has
\begin{align*}
&amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}(\ell+k+r)
!x^{n+1+r-k}\sigma _{\ell+k}(x) \\
&amp;+(-1)^{r+1}\sum_{k=0}^{\ell +r}(-1)^{k}\binom{\ell+r}{k}(
n+k+r)!x^{\ell +1+r-k}\sigma _{n+k}(x)=0.
\end{align*}
For $r$ odd and $n=\ell$, one obtains
\begin{equation*}
\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}(n+k+r)!x^{n+r-k}\sigma
_{n+k}(x)=0.
\end{equation*}

\section{Identities for other polynomials}

In the previous paragraphs, we have shown that Theorem \ref{theo}
generalizes several well-known identities involving Bernoulli numbers and
Bernoulli polynomials proven by numerous authors through various methods. The following
theorem is useful because similarly to Theorem \ref{theo}, it allows us
to establish analogous identities for particular sequences such as Fibonacci
and Lucas polynomial sequences.
\begin{theorem}\label{theorem2} Let $(w_{n})_{n\geq 0}$ be a sequence such that exponential generating
function $S_{w}(t)=\sum_{n=0}^{\infty}w_{n}\frac{t^{n}}{n!}$
satisfies $S_{w}(t) =\varepsilon e^{\lambda t}S_{w}(-t)$
where $\varepsilon =\pm 1$ and $\lambda \in \mathbb{C}$. Then for all non-negative integers $n$, $\ell$, and $r$, we have
      \begin{align}
         &amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
            \lambda ^{n+r-k}w_{\ell +k}\text{ }\notag \\
         &amp;+(-1)^{r+1}\varepsilon
            \sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}\binom{n+k+r}{r}
            \lambda ^{\ell+r-k}w_{n+k}=0.   \label{chell}
      \end{align}
\end{theorem}

\begin{proof}
Let $\mathcal{L}_{w}$ be a linear transformation defined over
polynomial space in $x$ specifying its action on monomials as
follows:
\begin{equation*} \mathcal{L}_{w}\left(x^{k}\right)=w_{k}.
\end{equation*} The relation $S_{w}(t)=\varepsilon S_{w}(-t)$
is equivalent to \begin{equation} w_{n}=\varepsilon\sum
(-1)^{k}\binom{n}{k}\lambda^{n-k}w_{k}, n\geq 0,  \label{cha1}
\end{equation} which can be written as \begin{equation*} x^{n}-\varepsilon
(\lambda-x)^{n}\in \ker \mathcal{L}_{w},\ n\geq 0.  \end{equation*} By
linearity, any polynomial $P(x)$, by the previous conditions satisfies
\begin{equation*} P(x)-\varepsilon P(\lambda-x) \in \ker\mathcal{L}_{w}.
\end{equation*} 
Let us consider $P(x)=x^{\ell+r}(\lambda-x)^{n+r}$
and note that $P(\lambda-x)=x^{n+r}(\lambda-x)^{\ell+r}$. Then
one obtains \begin{equation*}
\frac{D^{r}}{r!}\bigl(P(x)\bigl)=\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}\lambda^{n+r-k}x^{\ell+k}
\end{equation*} and \begin{equation*}
\frac{D^{r}}{r!}\bigl(P(\lambda-x)\bigl)=\sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}\binom{n+k+r}{r}\lambda
^{\ell+r-k}x^{n+k}.  \end{equation*} Thus, by combining the previous
formulae, we find Relation (\ref{chell}) of Theorem \ref{theorem2}.
\end{proof}

\subsection{Identities for Fibonacci and Lucas polynomials}

The bivariate polynomials of Fibonacci and Lucas $\bigl(u_{n}(
x,y)\bigl)_{n\geq 0}$ and $\bigl(v_{n}(x,y)\bigl)_{n\geq 0}$
are defined in \cite{bel1} by
\begin{equation*}
u_{n}(x,y)=xu_{n-1}(x,y)+yu_{n-2}(x,y)
\text{ and }v_{n}(x,y)=xv_{n-1}(x,y)
+yv_{n-2}(x,y)
\end{equation*}
for $n\geq 2$ with $u_{0}(x,y)=0$, $u_{1}(x,y)=1$,
$v_{0}(x,y)=2$, $v_{1}(x,y)=x$.
It is well-known that we have
\begin{equation*}
u_{n}(x,y)=\frac{\bigl{(\alpha(x,y)\bigl)^{n}-\bigl(\beta (x,y)\bigl)^{n}}}{\alpha\left(x,y\right)
-\beta\left(x,y\right)},\ v_{n}(x,y)=\bigl(\alpha
(x,y)\bigl)^{n}+\bigl(\beta(x,y)\bigl)^{n}
\end{equation*}
where
\begin{equation*}
\alpha(x,y)=\frac{1}{2}\bigg(x+\sqrt{x^{2}+4y}\bigg) \text{
and }\beta(x,y)=\frac{1}{2}\bigg(x-\sqrt{x^{2}+4y}\bigg).
\end{equation*}
Let us consider the sequences $u^{\ast}=\bigl(u_{sn}(x,y)\bigl)_{n}$ and $v^{\ast}=\bigl(v_{sn}(x,y)\bigl)_{n}$
where $s$ is a non-negative integer. This implies that
\begin{equation*}
S_{u^{\ast}}(t)=-e^{v_{s}(x,y)t}S_{u^{\ast
}}(-t)\text{ and }S_{v^{\ast}}(t)=e^{v_{s}(
x,y) t}S_{v^{\ast}}(-t).
\end{equation*}
Applying Theorem \ref{theorem2} leads to the following corollary:
\begin{corollary}\label{dercor}Let $n$, $\ell$, $r$, and $s$ be non-negative integers. Then
      \begin{align}
          &amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
           \bigl(v_{s}(x,y)\bigl)^{n+r-k}u_{s}(l+k)(x,y) \notag \\
           &amp;+(-1)^{r}\sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}\binom{
            n+k+r}{r}\bigl(v_{s}(x,y)\bigl)^{\ell+r-k}u_{s(n+k)}(x,y)=0, \\
          &amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
        \bigl(v_{s}(x,y)\bigl)^{n+r-k}v_{s}(l+k)(x,y) \notag \\
        &amp;-(-1)^{r}\sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}\binom{
        n+k+r}{r}\bigl(v_{s}(x,y)\bigl)^{\ell+r-k}v_{s(n+k)}(x,y)=0.
      \end{align}
\end{corollary}
Corollary \ref{dercor} generalizes identities involving Fibonacci and Lucas
numbers \cite{kos1} mentioned in \cite{ben1} and \cite{chu}. As it provides
identities for several sequences of integers, such as the Fibonacci sequence
$(F_{n})_{n\geq 0}=(u_{n}(1,1))_{n\geq 0}$, the
Lucas sequence $(L_{n})_{n\geq 0}=(v_{n}(1,1))_{n\geq 0}$,
the sequence of Pell numbers $(P_{n})_{n\geq 0}=(
u_{n}(2,1))_{n\geq 0}$, the sequence of Pell-Lucas numbers
$\left(Q_{n}\right)_{n\geq 0}=(v_{n}(2,1))_{n\geq 0}$, the
sequence of Jacobsthal numbers $(J_{n})_{n\geq 0}=(v_{n}(
1,2))_{n\geq 0}$, and the sequence Jacobsthal-Lucas numbers
$(j_{n})_{n\geq 0}=(v_{n}(1,2))_{n\geq 0}$ appear respectively in
the OEIS ({\it On-Line Encyclopedia of Integer Sequences})
\cite{slo} as \seqnum{A000045}, \seqnum{A000032}, \seqnum{A000129}, \seqnum{A002203}, \seqnum{
A001045} and \seqnum{A014551} and also the sequences $\left(F_{2n}\right)_{n\geq 0}$,
$\left(L_{2n}\right)_{n\geq 0}$, $(P_{2n})_{n\geq 0}$, $(Q_{2n})_{n\geq 0}$,
$(J_{2n})_{n\geq 0}$, and $(j_{2n})_{n\geq 0}$ respectively as \seqnum{A001906}, \seqnum{A005248},
\seqnum{A001542}, \seqnum{A003499}, \seqnum{A002450}, and \seqnum{A052539} in the OEIS.

\subsection{Identities for Chebyshev polynomials}

We can also apply Theorem \ref{theorem2} to Chebyshev
polynomials of the first kind $T(x)=(T_{n}(x))_{n\geq 0}$
and to Chebyshev polynomials of the second kind $U(x)=(U_{n}(x))_{n\geq 0}$
defined recursively in \cite{kos2,bel2} by
\begin{equation*}
T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \text{ and }
U_{n}(x)=2xU_{n-1}(x)-U_{n-2}(x)
\end{equation*}
for $n\geq 2$ with $T_{0}(x)=1$, $T_{1}(x)=x$, $
U_{0}(x)=1$, and $U_{1}(x)=2x.$
Clearly
\begin{equation*}
T_{n}(x)=\frac{1}{2}v_{n}(2x,-1)\text{ \ and }
U_{n}(x)=u_{n+1}(2x,-1).
\end{equation*}
By applying Corollary \ref{dercor}, we deduce the following relations:
\begin{align}
&amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
(2T_{s}(x))^{n+r-k}T_{s}(\ell+k)(x) \notag \\
&amp;-(-1)^{r}\sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}\binom{n+k+r}{r}
(2T_{s}(x))^{\ell+r-k}T_{s}(n+k)(x)  \notag \\
&amp;=0.
\end{align}
\begin{align}
&amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
\bigl(2T_{s}(x)\bigl)^{n+r-k}U_{s(\ell +k)
-1}\left(x\right)  \notag \\
&amp;+(-1)^{r}\sum_{k=0}^{\ell +r}(-1)^{k}\binom{\ell+r}{k}\binom{
n+k+r}{r}\bigl(2T_{s}(x)\bigl)^{\ell+r-k}U_{s(
n+k)-1}(x) \notag\\
&amp;=0,
\end{align}
where $n,\ell$, and $s$ are positive integers.

\subsection{Identities for generalized Euler polynomials}

Generalized Euler polynomials $E_{n}^{(\alpha)}(x)$ \cite[p. 102]{rom} are defined,
for $\alpha \in \mathbb{C}$, by
\begin{equation*}
S_{E^{(\alpha)}}(t)=\sum_{n=0}^{\infty
}E_{n}^{(\alpha)}(x)\frac{t^{n}}{n!}=\bigg(\frac{
2}{e^{t}+1}\bigg)^{\alpha }e^{tx}.
\end{equation*}
One has
\begin{equation*}
S_{E^{(\alpha)}}(t)=e^{(2x-\alpha)
t}S_{E^{(\alpha)}}(-t).
\end{equation*}
By applying Theorem \ref{theorem2}, one obtains the following identity:
\begin{align}
&amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
(2x-\alpha)^{n+r-k}E_{\ell +k}^{(\alpha)}(x) \notag \\
&amp;+(-1)^{r+1}\sum_{k=0}^{\ell +r}(-1)^{k}\binom{\ell+r}{k}
\binom{n+k+r}{r}(2x-\alpha)^{\ell+r-k}E_{n+k}^{(\alpha)}(x) \notag \\
&amp;=0.
\end{align}

\subsection{Identity for generalized Genocchi polynomials}

Generalized Genocchi polynomials $G_{n}^{(m)}(x)$ are defined, for $m\in \mathbb{N}$, by
\begin{equation*}
\sum_{n=0}^{\infty }G_{n}^{(m)}(x)\frac{t^{n}}{n!}
=\bigg(\frac{2t}{e^{t}+1}\bigg)^{m}e^{tx}.
\end{equation*}
One has
\begin{equation*}
S_{G^{(m)}}(t)=(-1)^{m}e^{(2x-m)t}S_{E^{(\alpha)}}(-t).
\end{equation*}
Theorem \ref{theorem2} leads to
\begin{align}
&amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
(2x-m)^{n+r-k}G_{\ell+l}^{(m)}(x) \notag\\
&amp;+(-1)^{m+r+1}\sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}
\binom{n+k+r}{r}(2x-m)^{\ell+r-k}G_{n+k}^{(m)}(x) \notag \\
&amp;=0.
\end{align}

\subsection{Identities for generalized Stirling numbers of the second kind}

The Stirling numbers of the second kind appear in the OEIS as \seqnum{A008277}, are defined by the explicit formula \cite{gra}
\begin{equation*}
S(n,m)=\frac{(-1)^{m}}{m!}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}j^{n}.
\end{equation*}
The first generalization of these numbers was provided by d'Ocagne \cite{mdo} and Carlitz \cite{car2}:
\begin{equation*}
S^{(\alpha)}(n,m)=\frac{(-1)^{m}}{m!}
\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}(\alpha+j)^{n}.
\end{equation*}
These numbers are connected with generalized Bernoulli polynomials \cite{ben2}:
\begin{equation*}
S^{(\alpha)}(n+m,m)=\binom{n+m}{m}B_{n}^{(-m)}(\alpha).
\end{equation*}
We are interested in the generalized Stirling numbers \cite{cak},
which are defined by
\begin{equation*}
S^{(\alpha)}(n,m,s)=\frac{(-1)^{m}}{m!}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}(\alpha +sj)^{n}.
\end{equation*}
Note that
\begin{equation*}
S^{(\alpha)}(n,m,1)=S^{(\alpha)}(n,m)\text{ \ and }S^{(0)}(n,m,1)
=S(n,m).
\end{equation*}
For $m$ and $s$ fixed, let us consider the sequence
$(w_{m,s})_{n}=(S^{(\alpha)}(n,m,s))_{n},$ for which the generating function is
\begin{equation*}
\sum_{n=0}^{\infty}S^{(\alpha)}(n,m,s)\frac{t^{n}
}{n!}=\frac{1}{m!}e^{\alpha t}(e^{st}-1)^{m}.
\end{equation*}
One has
\begin{equation*}
S_{w_{m,s}}(t)=(-1)^{m}e^{(2x+ms)t}S_{w_{m,s}}(-t).
\end{equation*}
Applying Theorem \ref{theorem2}, we obtain the following identity:
\begin{align}
&amp;\sum_{k=0}^{n+r}(-1)^{k}\binom{n+r}{k}\binom{\ell+k+r}{r}
(2x+ms)^{n+r-k}S^{(\alpha)}(\ell+k,m,s) \notag\\
&amp;+(-1)^{m+r+1}\sum_{k=0}^{\ell+r}(-1)^{k}\binom{\ell+r}{k}
\binom{n+k+r}{r}(2x+ms)^{\ell+r-k}S^{(\alpha)}(n+k,m,s) \notag\\
&amp;=0.
\end{align}

\section{Conclusion}
The study permits the development of two important results. The first one
proposes an identity allowing the generalization of various relations
related to classical Bernoulli numbers and Bernoulli polynomials, in addition to
Gessel's recent identity. The second result allows the deduction of similar
identities for Fibonacci, Lucas, and Chebyshev polynomials as well as for
the generalized Euler, Genocchi polynomials, and generalized Stirling numbers.

\section{Acknowledgments}
The authors would like to thank the referee and the editor for their valuable comments and
suggestions.

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\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}: Primary 11B68;
Secondary 05A10, 11B65.

\noindent \emph{Keywords:} Bernoulli polynomial, Bernoulli number, identity.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000032},
\seqnum{A000045},
\seqnum{A000129},
\seqnum{A001045},
\seqnum{A001542},
\seqnum{A001906},
\seqnum{A002203},
\seqnum{A002450},
\seqnum{A003499},
\seqnum{A005248},
\seqnum{A008277},
\seqnum{A014551}, and
\seqnum{A052539}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received June 30 2020;
revised versions received  July 19 2020; 
October 14 2020; November 16 2020; November 18 2020.
Published in {\it Journal of Integer Sequences},
November 24 2020.

\bigskip
\hrule
\bigskip

\noindent
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\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
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