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\begin{center}
\vskip 1cm{\LARGE\bf
Fourier Expansions and Integral\\
\vskip .1in
Representations for Genocchi Polynomials}
\vskip 1cm
\large
Qiu-Ming Luo\\
Department of Mathematics\\
East China Normal University\\
Dongchuan Road 500 \\
Shanghai City, 200241\\
China\\
and \\
Department of Mathematics\\
Jiaozuo University \\
Jiaozuo City, Henan Province, 454003\\
China\\
\href{mailto:luomath2007@163.com}{\tt luomath2007@163.com}
\end{center}

\vskip .2in

\begin{abstract}
In this paper, by using the Lipschitz summation formula, we obtain
Fourier expansions and integral representations for the Genocchi
polynomials. Some other new and interesting results are also shown.
\end{abstract}

\section{Introduction}

It is well-known that Genocchi polynomials $G_{n}(x)$ are defined \cite{Sandor} by
\begin{equation}\label{A32}
\frac{2ze^{xz}}{e^{z}+1}=\sum_{n=0}^{\infty}G_{n}(x) \frac{z^n}{n!},\quad \vert z\vert <\pi.
\end{equation}
In particular, the quantities $G_n\triangleq G_n(0)$ for $n\ge0$ are called Genocchi numbers, with $G_{2n+1}=0$ for $n\geq1$ and, for example,
\begin{align*}
G_0&=0, & G_1&=1, & G_2&=-1, & G_4&=1, & G_6&=-3, & G_8&=17, & G_{10}&=-155, & G_{12}&=2073.
\end{align*}
This is Sloane's sequence \seqnum{A001469}.

The $n$-th Genocchi function $\widehat{{G}}_n(x)$ may be introduced in the following way: for $0\leq x<1$ and $n\ge0$,
\begin{equation}\label{A991}
\widehat{{G}}_n(x)\triangleq{G}_n(x) \quad \text{and}\quad \widehat{{G}}_n(x+1)=-\widehat{{G}}_n(x);
\end{equation}
for $x \in \mathbb{R}$ and $r \in \mathbb{Z}$,
\begin{equation}\label{A992}
\widehat{{G}}_n(x) = (-1)^{\lfloor x \rfloor}{G}_n(\{x\}) \quad\text{and}\quad \widehat{{G}}_n(x+r)=(-1)^r \widehat{{G}}_n(x),
\end{equation}
where the symbols $\{x\}$ and $\lfloor x \rfloor$ denote the fractional part of $x$ and the greatest integer not exceeding $x$ respectively. Sometimes we also call $\widehat{{G}}_n(x)$ the periodic Genocchi polynomials. For convenience, in what follows, we would still employ ${G}_n(x)$ to stand for the periodic Genocchi polynomials, when no confusion appears in the context.
\par
It is also well-known that Euler polynomials $E_{n}(x)$ for $n\ge0$ may be defined \cite{abram, Comtet1974} by
\begin{equation}\label{A31}
\frac{2e^{xz}}{e^{z}+1}=\sum_{n=0}^{\infty}E_{n}(x) \frac{z^n}{n!},\quad \vert z\vert <\pi.
\end{equation}
By~\eqref{A32} and~\eqref{A31}, we can obtain the following relation between Euler polynomials $E_{n}(x)$ and Genocchi polynomials $G_{n}(x)$:
\begin{equation}\label{relat}
G_{n}(x)=nE_{n-1}(x),
\end{equation}
or
\begin{equation}\label{relat2}
E_{n}(x)= \frac{1}{n+1}G_{n+1}(x).
\end{equation}
\par
In this paper, by using the Lipschitz summation formula, we establish Fourier expansions and integral representations for Genocchi polynomials and present an explicit formula for Genocchi polynomials at rational arguments.

\section{Fourier expansions for Genocchi polynomials}

Recall \cite{Lipschitz1889} that the Lipschitz summation formula states that
\begin{equation}\label{A1050}
\sum_{n+\mu>0}\frac{e^{2{\pi}i(n+\mu)\tau}}{(n+\mu)^{1-\alpha}} =\frac{\Gamma(\alpha)}{(-2{\pi}i)^\alpha}\sum_{k \in \mathbb{Z}} \frac{e^{-2{\pi}ik{\mu}}}{(\tau+k)^\alpha},
\end{equation}
where $\alpha \in \mathbb{C}$, either $\Re(\alpha) > 1$ for $\mu \in \mathbb{Z}$ or $\Re(\alpha)>0$ for $\mu \in \mathbb{R} \setminus \mathbb{Z}$, $\tau$ belongs to the upper half of the complex plane, and $\Gamma$ is Euler gamma function.
\par
In virtue of the Lipschitz summation formula~\eqref{A1050}, we obtain Fourier expansions for Genocchi polynomials as follows.

\begin{theorem} \label{A1210}
For either $n=0$ and $0<x<1$ or $n>0$ and $0\leq x\leq1$,
\begin{align}\label{A1211}
G_n(x)& =\frac{2\cdot n!}{({\pi}i)^n} \sum_{k \in \mathbb{Z}} \frac{e^{(2k-1){\pi}ix}}{(2k-1)^n}\\
\label{A1212}
& =\frac{4 \cdot n!}{\pi^n} \sum_{k=0}^{\infty}\frac{\cos[(2k+1)\pi{x}-{n\pi}/{2}]}{(2k+1)^n}.
\end{align}
\end{theorem}

\begin{proof}
For $0 \leq x \leq 1$, utilization of~\eqref{A32} and the generalized binomial theorem yields
\begin{equation}\label{A1213}
\sum_{k=0}^{\infty}G_{k}(x) \frac{(2{\pi}i\tau)^{k-1}}{k!}
=\frac{2e^{2{\pi}i{\tau}x}}{e^{2{\pi}i\tau}+1}
=2\sum_{k=0}^{\infty}(-1)^ke^{2{\pi}i(k+x)\tau}, \quad \vert\tau\vert<\frac12.
\end{equation}
Differentiating $n-1$ times with respect to the variable $\tau$ on both sides of~\eqref{A1213} gives
\begin{equation}\label{A1214}
\sum_{k=n}^{\infty}G_{k}(x) \frac{(2{\pi}i)^{k-1}\tau^{k-n}}{k(k-n)!}=2(2{\pi}i)^{n-1} \sum_{k=0}^{\infty}(-1)^k(k+x)^{n-1}e^{2{\pi}i(k+x)\tau}.
\end{equation}
On the other hand, replacing $\tau$ by $\tau+\frac12$ and letting $\alpha=n$ and $\mu=x$ in Lipschitz summation formula~\eqref{A1050} lead to
\begin{equation}\label{A1215}
\frac{(n-1)!}{(-{\pi}i)^n}\sum_{k \in \mathbb{Z}} \frac{e^{-(2k+1){\pi}ix}}{(2k+2\tau+1)^n}=\sum_{k=0}^{\infty}(-1)^k(k+x)^{n-1}e^{2{\pi}i(k+x)\tau}.
\end{equation}
Combining~\eqref{A1214} and~\eqref{A1215} reveals
\begin{equation}\label{A1216}
\sum_{k=n}^{\infty
}G_{k}(x) \frac{(2{\pi}i)^{k-1}\tau^{k-n}}{k(k-n)!}=\frac{(-1)^{n}2^n(n-1)!}{{\pi}i}\sum_{k \in \mathbb{Z}} \frac{e^{-(2k+1){\pi}ix}}{(2k+2\tau+1)^n}.
\end{equation}
Taking $\tau \to 0$ in~\eqref{A1216} gives the equation~\eqref{A1211}.
\par
Since $i^{-n}=e^{-\frac{n{\pi}i}{2}}$, the equation~\eqref{A1212} follows as a direct consequence of~\eqref{A1211}.
\end{proof}

The following corollary is a straightforward consequence of Theorem~\ref{A1210}.

\begin{corollary}\label{A1212w}
For either $n=0$ and $0<x<1$ or $n>0$ and $0\leq x\leq1$,
\begin{equation}
G_{2n}(x)=(-1)^n\frac{4 \cdot (2n)!}{{\pi}^{2n}} \sum_{k=0}^{\infty} \frac{\cos[(2k+1)\pi{x}]}{(2k+1)^{2n}}
\end{equation}
and
\begin{equation}
G_{2n-1}(x)=(-1)^{n-1}\frac{4 \cdot (2n-1)!}{{\pi}^{2n-1}} \sum_{k=0}^{\infty} \frac{\sin[(2k+1)\pi{x}]}{(2k+1)^{2n-1}}.
\end{equation}
\end{corollary}

\begin{remark}
From Fourier expansions of Euler polynomials (see \cite{abram, Cvijovic1999, Magnus}) and the equation~\eqref{relat}, We can directly derive the formula~\eqref{A1212} and Corollary~\ref{A1212w}. Conversely, we can also recover some known Fourier expansions of Euler polynomials by applying the relation~\eqref{relat2}, Theorem~\ref{A1210} and Corollary~\ref{A1212w}.
\end{remark}

\section{Integral representations for Genocchi polynomials}

Now we are in a position to state and prove the uniform integral representations for Genocchi polynomials as follows.

\begin{theorem} \label{A1217}
For $n\in\mathbb{N}$ and $0 \leq \Re(x) \leq 1$,
\begin{align}\label{A1218}
G_n(x) =2 n \int_0^\infty \frac{e^{{\pi}t}\cos(\pi{x}-n\pi/2)-e^{-\pi{t}}\cos(\pi{x}+n\pi/2)} {\cosh(2{\pi}t)-\cos(2{\pi}x)}t^{n-1}\td{t}.
\end{align}
\end{theorem}

\begin{proof}
Utilizing
\begin{equation*}
G_n(x) =\frac{4 \cdot n!}{{\pi}^n} \sum_{k=0}^{\infty} \frac{\cos[(2k+1)\pi{x} -n\pi/2]}{(2k+1)^n}
\end{equation*}
and
\begin{align}\label{A1070}
\int_0^\infty t^ne^{-at}\td{t}=\frac{n!}{a^{n+1}}
\end{align}
for $n\ge0$ and $\Re(a)>0$ reveals that
\begin{align*}
G_n(x) & =\frac{4 n}{{\pi}^n} \sum_{k=0}^{\infty}\cos\biggl[(2k+1)\pi{x}-\frac{n\pi}{2}\biggr] \int_0^{\infty}t^{n-1}e^{-(2k+1)t}\td{t}\\
& =\frac{4 n}{{\pi}^n} \int_0^{\infty} t^{n-1} \sum_{k=0}^{\infty}e^{-(2k+1)t}\cos\biggl[(2k+1)\pi{x}-\frac{n\pi}{2}\biggr]\td{t}\\
&=\frac{4 n}{{\pi}^n} \int_0^\infty \Biggl\{\cos\biggl(\frac{n\pi}{2}\biggr) \sum_{k=0}^{\infty} e^{-(2k+1)t}\cos [(2k+1)\pi{x}]\\
& \quad+ \sin \biggl(\frac{n\pi}{2}\biggr) \sum_{k=0}^{\infty} e^{-(2k+1)t}\sin [(2k+1)\pi{x}]\Biggr\}t^{n-1}\td{t}.
\end{align*}
By making use of
\begin{equation*}
\sum_{k=0}^{\infty} e^{-(2k+1)t} \sin[(2k+1)x]=\frac{\sin{x}\cosh{t}}{\cosh(2t)-\cos(2x)}
\end{equation*}
and
\begin{equation}
\sum_{k=0}^{\infty} e^{-(2k+1)t} \cos[(2k+1)x]=\frac{\cos{x}\sinh{t}}{\cosh(2t)-\cos(2x)}
\end{equation}
for $t>0$, which may be deduced from
\begin{equation*}
\sum_{k=0}^{\infty} e^{(xi-t)(2k+1)}=\frac{\cos{x}\sinh{t}+i\sin{x}\cosh{t}}{\cosh(2t)-\cos(2x)}
\end{equation*}
for $t>0$, and applying the transformation $t= {\pi}s$, the desired formula~\eqref{A1218} follows.
\end{proof}

It is easy to see that Theorem~\ref{A1217} implies the following integral representations for Genocchi polynomials.

\begin{corollary}
For $n\in\mathbb{N}$ and $0 \leq \Re(x) \leq 1$,
\begin{align}\label{A1219}
G_{2n-1}(x) &=4(2n-1)(-1)^{n-1}\int_0^\infty \frac{\sin(\pi{x})\cosh(\pi{t})}{\cosh(2{\pi}t)-\cos(2{\pi}x)}t^{2n-2}\td{t}
\end{align}
and
\begin{align}
\label{A1220}
G_{2n}(x) &=8n(-1)^n \int_0^\infty \frac{\cos(\pi{x})\sinh({\pi}t)}{\cosh(2{\pi}t)-\cos(2{\pi}x)}t^{2n-1}\td{t}.
\end{align}
\end{corollary}

\begin{remark}
The uniform integral representations for Genocchi polynomials are not found in the classical literatures such as \cite{abram, ErdelyiI, Magnus}. So the formula~\eqref{A1218} is presumably new.
\end{remark}

\begin{remark}
Our method used in this section can also be applied to establish uniform Fourier expansions and uniform integral representations for both Bernoulli and Euler polynomials.
\end{remark}

\begin{remark}
Please note that Theorem~\ref{A1210} can be derived from Theorem~\ref{A1217}.
\end{remark}

\section{Corollaries of uniform integral representations}

Finally, we present some corollaries of Theorem~\ref{A1217}.

\begin{corollary} \label{A1221}
For $n\in\mathbb{N}$ and $0 \leq \Re(x) \leq 1$,
\begin{align}\label{A1222}
G_n(x) =(-1)^{n-1}\frac{4n}{\pi^n} \int_0^1 \frac{\cos(\pi{x}-n\pi/2)-t^2 \cos(\pi{x}+n\pi/2)}{t^4-2{t^2}\cos(2{\pi}x)+1}(\log{t})^{n-1}\td{t}.
\end{align}
\end{corollary}
\begin{proof}
Substituting
$$
\cosh(2{\pi}t)=\frac{e^{2{\pi}t}+e^{-2{\pi}t}}{2}
$$
into~\eqref{A1218} gives
\begin{align}\label{A1223}
G_n(x) =4 n \int_0^\infty \frac{e^{{\pi}t}\cos(\pi{x}-n\pi/2)-e^{-\pi{t}}\cos(\pi{x}+n\pi/2)} {e^{2{\pi}t}+e^{-2{\pi}t}-2\cos(2{\pi}x)}t^{n-1}\td{t}.
\end{align}
Further carrying out the transformation $u=e^{-{\pi}t}$ in~\eqref{A1223} yields the desired formula~\eqref{A1222}.
\end{proof}

It is easy to see that the following formulas can be deduced from Corollary~\ref{A1221}.

\begin{corollary}\label{A1225}
For $n\in\mathbb{N}$ and $0 \leq \Re(x) \leq 1$,
\begin{align}\label{A1226}
G_{2n-1}(x) &=(-1)^{n-1}\frac{4(2n-1)}{\pi^{2n-1}} \int_0^1 \frac{(1+t^2) \sin(\pi{x})}{t^4-2{t^2}\cos(2{\pi}x)+1}(\log{t})^{2n-2}\td{t}
\end{align}
and
\begin{align}\label{A1227}
G_{2n}(x) &=(-1)^{n-1}\frac{8n}{\pi^{2n}} \int_0^1 \frac{(1-t^2) \cos(\pi{x})}{t^4-2{t^2}\cos(2{\pi}x)+1}(\log{t})^{2n-1}\td{t}.
\end{align}
\end{corollary}

In \cite[p. 35, (21)]{ErdelyiI}, it was listed that
\begin{align}\label{A30sop}
\zeta(2n)=\frac{(-1)^{n-1}(2\pi)^{2n}}{2(2n)!}B_{2n},
\end{align}
where $\zeta (s)$ is the Riemann zeta function defined by
\begin{equation}\label{A122}
\zeta (s)=\sum_{n=1}^{\infty }\frac{1} {n^{s}},\quad s>0
\end{equation}
and $B_{n}$ for $n\ge0$ are Bernoulli numbers defined by
\begin{equation}\label{A30s}
\frac{z}{e^{z}-1}=\sum_{n=0}^{\infty}B_{n} \frac{z^n}{n!},\quad \vert z\vert <2\pi.
\end{equation}
By~\eqref{A30sop} and
\begin{equation}\label{A30sx}
G_{2n}=2(1-2^{2n})B_{2n},
\end{equation}
it follows that
\begin{align}\label{A1228}
G_{2n}=\frac{(-1)^{n-1}2^{2}(1-2^{2n})(2n)!}{(2\pi)^{2n}}\zeta(2n).
\end{align}
Combining the formula
$$
\int_a^xG_{n}(t)\td{t}=\frac{G_{n+1}(x)-G_{n+1}(a)}{n+1}
$$
in~\cite{Sandor} and the integral formulas
\begin{align*}
\int \frac{2t(1-t^2)\cos{{x}}}{t^4-2{t^2}\cos(2x)+1}\td{x}&=\arctan \biggl(\frac{2t \sin{x}}{1-t^2}\biggr)+C,\\*
\int \frac{4t(1+t^2)\sin{{x}}}{t^4-2{t^2}\cos(2x)+1}\td{x}&=\log \biggl(\frac{t^2-2t \cos{x}+1}{t^2+2t \cos{x}+1}\biggr)+C,\\*
\int _0^1\frac{\log(1+t)(\log{t})^{n-1}}{t}\td{t} &=(-1)^{n}(n-1)!\biggl(\frac1{2^{n}}-1\biggr)\zeta(n+1),\\*
\int _0^1\frac{\log(1-t)(\log{t})^{n-1}}{t}\td{t}&=(-1)^{n}(n-1)!\zeta(n+1)
\end{align*}
in~\cite{Prudnikov} with~\eqref{A1228} and Corollary~\ref{A1225} gives the following corollary.

\begin{corollary}\label{A1229}
For $n\in\mathbb{N}$ and $0 \leq \Re(x) \leq 1$,
\begin{align}
\label{A1230}
G_{2n+1}(x)&=(-1)^{n-1}\frac{4n(2n+1)}{\pi^{2n+1}} \int_0^1 \arctan\biggl[\frac{2t\sin({\pi}x)}{1-t^2}\biggr] \frac{(\log{t})^{2n-1}}{t}\td{t}
\end{align}
and
\begin{align}\label{A1231}
G_{2n}(x)&=(-1)^{n-1}\frac{2n(2n-1)}{\pi^{2n}} \int_0^1 \log \biggl[\frac{t^2-2t \cos({\pi}x)+1}{t^2+2t \cos({\pi}x)+1}\biggr]\frac{(\log{t})^{2n-2}}{t}\td{t}.
\end{align}
\end{corollary}

By~\eqref{A1219},~\eqref{A1227} and~\eqref{A1231}, the following integral representations for Genocchi numbers can be obtained.

\begin{corollary} \label{A1232}
For $n\geq0$,
\begin{align*}
G_{2n} &= (-1)^n4n \int_0^\infty t^{2n-1}\textup{csch}({\pi}t)\td{t}\\
&=(-1)^{n+1}\frac{8n}{\pi^{2n}}\int_0^1 \frac{(\log{t})^{2n-1}}{1-t^2}\td{t}\\
&= (-1)^{n-1}\frac{4n(2n-1)}{\pi^{2n}} \int_0^1 \log \biggl(\frac{1-t}{1+t}\biggr)\frac{(\log{t})^{2n-2}}{t}\td{t}.
\end{align*}
\end{corollary}

Finally, we give an explicit formula for Genocchi polynomials at rational arguments.

\begin{theorem}\label{A245}
For $n,q\in\mathbb{N}$ and $p\in\mathbb{Z}$,
\begin{equation}
\begin{split}\label{A245v}
G_{n}\biggl( \frac{p}{q}\biggr) =\frac{4\cdot n!}{(2q\pi)^n}
\sum_{j=1}^{q}\zeta \biggl( n,\frac{2j-1}{2q}\biggr)
\cos \biggl[\frac{(2j-1)p \pi}{q} - \frac{n\pi}{2}\biggr],
\end{split}
\end{equation}
where
\begin{equation}\label{A53}
\zeta (s,a)=\sum_{n=0}^{\infty }\frac{1}{(n+a)^{s}}
\end{equation}
for $\mathfrak{R}(s)>1$ and $a\notin\mathbb{Z}_{0}^{-}$ is Hurwitz zeta function (see~\cite{ErdelyiI, Magnus}).
\end{theorem}

\begin{proof}
The formula~\eqref{A1212} can be rewritten as
\begin{align*}
{G}_{n}(x)=\frac{4 \cdot n!}{\pi^n} \sum_{k=1}^{\infty} \frac{\cos[{n\pi}/{2}-(2k-1){\pi}x]}{(2k-1)^n}.
\end{align*}
By~\eqref{A53} and the elementary
series identity
\begin{equation}\label{A126}
\sum_{k=1}^{\infty }f( k) =\sum_{j=1}^{q}\sum_{k=0}^{\infty}f( {q} k+j), \quad q \in \mathbb{N},
\end{equation}
we obtain the desired formula~\eqref{A245v} by setting $x=\frac{p}{q}$. This completes the proof.
\end{proof}

From Theorem~\ref{A245}, we can easily deduce the following corollary.

\begin{corollary}\label{A245d}
For $n, q \in \mathbb{N}$ and $p\in\mathbb{Z}$,
\begin{align}\label{A245dd}
G_{2n}\biggl( \frac{p}{q}\biggr)& = (-1)^n \frac{4 \cdot (2n)!}{(2q \pi)^{2n}}
\sum_{j=1}^{q}\zeta \biggl( 2n,\frac{2j-1}{2q}\biggr)
\cos \biggl[\frac{(2j-1)p \pi}{q}\biggr]
\end{align}
and
\begin{align}
\label{A245xz}
G_{2n-1}\biggl( \frac{p}{q}\biggr)& = (-1)^{n-1} \frac{4 \cdot (2n-1)!}{(2q \pi)^{2n-1}}
\sum_{j=1}^{q}\zeta \biggl(2n-1,\frac{2j-1}{2q}\biggr)\sin \biggl[\frac{(2j-1)p \pi}{q}\biggr].
\end{align}
\end{corollary}

\begin{remark}
We can directly obtain the formulas~\eqref{A245dd} and~\eqref{A245xz} by using the relation~\eqref{relat} and the formulas~(12a) and~(12b) in~\cite{Cvijovic1999}. Similarly, we can also derive the corresponding formula for Euler polynomials at rational arguments by applying the relation~\eqref{relat2}, Theorem~\ref{A245} and Corollary~\ref{A245d}.
\end{remark}

\section{Acknowledgements}
The author would like to thank Professor Jeffrey Shallit and Professor Sen-Lin Guo in Canada, Professor Feng Qi in China and the anonymous referees for their valuable comments and suggestions for improving the original version of this manuscript.
\par
The present investigation was supported, in part, by the \textit{PCSIRT Project of the Ministry of Education of China} under Grant IRT0621 \textup{and} \textit{Innovation Program of Shanghai Municipal Education Committee of China} under Grant 08ZZ24 and \textit{Henan Innovation Project For University Prominent Research Talents of China} under Grant 2007KYCX0021.

\begin{thebibliography}{99}

\bibitem{abram}
M. Abramowitz and I. A. Stegun (Ed.), \textit{Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}, National Bureau of Standards, Applied Mathematics Series \textbf{55}, Fourth Printing, 1965.

\bibitem{Comtet1974}
L. Comtet, \textit{Advanced Combinatorics: The Art of Finite and Infinite Expansions}, Translated from the French by J. W. Nienhuys, Reidel, 1974.

\bibitem{Cvijovic1999}
D. Cvijovi\'{c}, J. Klinowski, Values
of the Legndre chi and Hurwitz zeta functions at rational arguments, \textit{Math. Comput.} \textbf{68} (1999), 1623--1630.

\bibitem{ErdelyiI}
A. Erd\'{e}lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, \textit{Higher Transcendental Functions}, Volume I, McGraw-Hill, 1953.

\bibitem{Lipschitz1889}
R. Lipschitz, Untersuchung der Eigenschaften einer Gattung von unendlichen Reihen, \textit{J. Reine und Angew. Math.} \textbf{105} (1889), 127--156.

\bibitem{Magnus}
W. Magnus, F. Oberhettinger and R. P. Soni, \textit{Formulas and Theorems for the Special Functions of Mathematical Physics}, 3rd Edition, Springer, 1966.

\bibitem{Prudnikov}
A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev,  \textit{Integrals
and Series: Elementary Functions}, Volume I, Gordon and Breach, 1986.

\bibitem{Sandor}
J. S\'{a}ndor and B. Crstici, \textit{Handbook of Number Theory II}, Kluwer Academic Publishers, 2004.

\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B83; Secondary 42A16.

\noindent \emph{Keywords: } 
Genocchi numbers, Genocchi polynomials, Fourier expansion, integral
representation, Lipschitz summation formula.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequence
\seqnum{A001469}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received June 24 2008;
revised versions received August 29 2008; October 4 2008.
Published in {\it Journal of Integer Sequences}, December 14 2008.

\bigskip
\hrule
\bigskip

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