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\vskip 1cm{\LARGE\bf A Short Proof of Carlitz's \\
\vskip .11in
Bernoulli Number Identity
}
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\large
Helmut Prodinger\\
Department of Mathematical Sciences\\
Stellenbosch University\\
7602 Stellenbosch\\
South Africa\\
\href{mailto:hproding@sun.ac.za}{\tt hproding@sun.ac.za} \\
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\begin{abstract}
For an identity related to Bernoulli numbers, stated by Carlitz,
rediscovered and reproved by various researchers, an extremely short
and direct proof is provided which uses a bivariate exponential
function.
\end{abstract}

\section{Introduction and result}

The very recent paper \cite{GoQu14} deals with the remarkable identity 
\begin{equation*}
(-1)^m\sum_{k=0}^m\binom mk B_{n+k}=
(-1)^n\sum_{k=0}^n\binom nk B_{m+k}
\end{equation*}
involving the Bernoulli numbers $(B_n)$.
We learn that it originated as a problem of Carlitz \cite{Carlitz}, with
a solution by Shannon \cite{Shannon}, which uses induction. It was rediscovered
by Vassilev and Vassilev-Missana \cite{VassiMissi}.
The paper by Gould and Quaintance 
\cite{GoQu14} introduces and uses a binomial transform to prove it.

In this quick note, I would like to present a proof by (exponential) generating functions, which is perhaps the most direct argument. It goes like this:
\begin{align*}
F(z,x)&amp;:=\sum_{m\ge0}\frac{z^m}{m!}\sum_{n\ge0}\frac{x^n}{n!}(-1)^m\sum_{k=0}^m\binom mk B_{n+k}\\
&amp;=\sum_{n\ge0}\frac{x^n}{n!}\sum_{k\ge0}\frac{B_{n+k}}{k!}\sum_{m\ge k}(-z)^m\frac{1}{(m-k)!} \\
&amp;=\sum_{n\ge0}\frac{x^n}{n!}\sum_{k\ge0}\frac{B_{n+k}}{k!}(-z)^ke^{-z} \\
&amp;=e^{-z}\sum_{N\ge0}\frac{B_N}{N!}\sum_{k=0}^N\binom Nk(-z)^kx^{N-k}\\
&amp;=e^{-z}\sum_{N\ge0}\frac{B_N}{N!}(x-z)^N\\
&amp;=e^{-z}\frac{x-z}{e^{x-z}-1}=\frac{x-z}{e^{x}-e^z}=F(x,z).\\
\end{align*}
This symmetry proves the identity.\qed

\begin{thebibliography}{9}

\bibitem{Carlitz}
L. Carlitz.
\newblock Problem 795.
\newblock {\it Math. Mag.} {\bf 44} (1971), 107.

\bibitem{GoQu14}
H. W. Gould and J. Quaintance.
\newblock Bernoulli numbers and a new binomial
transform identity.
\newblock {\it J. Integer Sequences} {\bf 17}  (2014), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL17/Quaintance/quain3.html}{Article 14.2.2}.

\bibitem{Shannon}
A. G. Shannon.
\newblock Solution of Problem 795.
\newblock {\it Math. Mag.} {\bf 45} (1972), 55--56.

\bibitem{VassiMissi}
P. Vassilev and M. Vassilev--Missana.
\newblock On one remarkable identity involving Bernoulli
numbers.
\newblock {\it Notes on Number Theory and Discrete Mathematics} {\bf 11} (2005), 22--24.



\end{thebibliography}


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

\noindent \emph{Keywords: } Bernoulli number, exponential generating function.

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\vspace*{+.1in}
\noindent
Received January 18 2014;
revised version received January 20 2014.
Published in {\it Journal of Integer Sequences}, February 15 2014.

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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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