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\begin{center}
\vskip 1cm{\LARGE\bf Carlitz's Identity for the Bernoulli Numbers \\
\vskip .1in
and Zeon Algebra} \vskip 1cm \large Ant\^onio Francisco
Neto\footnote{This work was supported by Conselho Nacional de
Desenvolvimento Cient\'ifico e Tecnol\'ogico
(CNPq-Brazil) under grant 307617/2012-2.}\\
DEPRO, Escola de Minas\\
Campus Morro do Cruzeiro, UFOP\\
35400-000 Ouro Preto MG \\
Brazil \\
\href{mailto:antfrannet@gmail.com}{\tt antfrannet@gmail.com}\\
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\begin{abstract}
In this work we provide a new short proof of Carlitz's
identity for the Bernoulli numbers. Our approach is based on the
ordinary generating function for the Bernoulli numbers and a
Grassmann-Berezin integral representation of the Bernoulli numbers
in the context of the Zeon algebra, which comprises an associative
and commutative algebra with nilpotent generators.
\end{abstract}

\section{Introduction}

In this work we will give a new, simple and short proof of 
Carlitz's identity for the Bernoulli numbers \cite{Carlitz}
\begin{equation}\label{CarBer}
\sum_{i=0}^m{m \choose i}B_{n+i}=(-1)^{m+n}\sum_{j=0}^n{n \choose j}B_{m+j},
\end{equation} using the Zeon algebra \cite{NetodAnjos,Neto}. The identity in Eq.~(\ref{CarBer}) has been re-obtained
many times \cite{Chen,Chu,Gessel,Shannon,VassiMissi} and also very
recently \cite{GoQu14,Prodinger,Singh}. The proof given here is of
independent interest, because of the simplicity of the arguments
involved and, as it has also occurred in other contexts
\cite{Abde,Bedi,Cara,Fein,Fein1,Mansour,NetodAnjos,Neto,Schor,Scho,Scho1},
the proof comprises yet another example of the usefulness of 
Zeon algebra and/or the Grassmann algebra in obtaining
combinatorial identities.

Before we continue, we establish the basic underlying algebraic
setup needed to give the proof of Eq.~(\ref{CarBer}). Throughout
this work we let $\mathbb{Q}$ and $\mathbb{R}$ denote the rational
and real numbers, respectively.

\section{Basic definitions: Zeon algebra and the Grassmann-Berezin integral}

\begin{definition}\label{Def1} The
\textit{Zeon algebra} $\mathcal{Z}_n \supset \mathbb{R}$ is
defined as the associative algebra generated by the collection
$\{\varepsilon_i\}_{i=1}^n$ ($n&lt;\infty$) and the scalar $1 \in
\mathbb{R}$, such that
$1\varepsilon_i=\varepsilon_i=\varepsilon_i1$, $\varepsilon_i
\varepsilon_j = \varepsilon_j \varepsilon_i$ $\forall$ $i$, $j$
and $\varepsilon_i^2=0$ $\forall$ $i$.
\end{definition} Note that
only linear elements in $\mathcal{Z}_n$ contribute to the
calculations.

For $\{i,j,\ldots,k\} \subset \{1,2,\ldots,n\}$ and
$\varepsilon_{ij\cdots k}\equiv \varepsilon_i\varepsilon_j\cdots
\varepsilon_k$ the most general element with $n$ generators
$\varepsilon_i$ can be written as (with the convention of sum over
repeated indices implicit)
\begin{equation}\label{phi}\phi_n=
a+a_i\varepsilon_i+a_{ij}\varepsilon_{ij}+\cdots+ a_{12\cdots
n}\varepsilon_{12\cdots n}=\sum_{\mathbf{i} \in
2^{[n]}}a_{\mathbf{i}}\varepsilon_\mathbf{i},\end{equation} with
$a$, $a_i$, $a_{ij}$, $\ldots$, $a_{12\cdots n}$ $\in$
$\mathbb{R}$, $2^{[n]}$ being the power set of
$[n]:=\{1,2,\ldots,n\}$, and $1\leq i&lt;j&lt; \cdots \leq n$. We refer
to $a$ as the body of $\phi_n$ and write $b(\phi_n)=a$ and to
$\phi_n-a$ as the soul such that $s(\phi_n)=\phi_n-a$.

\begin{definition}\label{Def2} The \textit{Grassmann-Berezin integral} on $\mathcal{Z}_n$, denoted
by $\int$, is the linear functional $\int: \mathcal{Z}_n
\rightarrow \mathbb{R}$ such that (we use throughout this work the
compact notation $d\mu_n:=d\varepsilon_n \cdots d\varepsilon_1$)
$$
d\varepsilon_id\varepsilon_j=d\varepsilon_jd\varepsilon_i,\,\,
\int \phi_n\bigl(\hat{\varepsilon}_i\bigr)d\varepsilon_i=0\,\,{\rm and}
\int \phi_n\bigl(\hat{\varepsilon}_i\bigr)\varepsilon_id\varepsilon_i=\phi_n\bigl(\hat{\varepsilon}_i\bigr),
$$ where $\phi_n\bigl(\hat{\varepsilon}_i\bigr)$ means any element of $\mathcal{Z}_n$ with no dependence on
$\varepsilon_i$. Multiple integrals are iterated integrals, i.e.,
$$
\int f(\phi_n) d\mu_n = \int \cdots \biggl(\int \biggl(\int f(\phi_n) d\varepsilon_n\biggr)
d \varepsilon_{n-1}\biggr)\cdots d\varepsilon_1.
$$\end{definition}
For example, if we define $\varphi_n:=
\varepsilon_1+\cdots+\varepsilon_n$ it follows directly from
Definition \ref{Def2} and the multinomial theorem that
\begin{equation}\label{IntAux}\int
\varphi_n^id\mu_n=i!\delta_{i,n},\end{equation} where $\delta_{i,n}$ is the Kronecker delta. For more details
on Grassmann-Berezin integration, we refer the reader to the books
of Berezin \cite[Chapter 1]{BreSQ} and \cite[Chapter 2]{BreIS} or
the books of DeWitt \cite[Chapter 1]{DeWitt} and Rogers
\cite[Chapter 11]{Rogers}.

We will now recall some basic facts about the Zeon algebra. First,
$a+\phi_n$ with $s(a)=0=b(\phi_n)$ is invertible iff $b(a)\neq 0$.
More precisely, we have
\begin{equation}\label{Inv}
\frac{1}{a+\phi_n}=\frac{1}{a}\left(1-\frac{\phi_n}{a}+\frac{\phi_n^2}{a^2}+\cdots+(-1)^n\frac{\phi_n^n}{a^n}\right).
\end{equation}
Second, the following expression holds
\begin{equation}\label{exp}
e^{\varphi_n}:=\sum_{i=0}^{\infty}\frac{\varphi_n^i}{i!}=\sum_{i=0}^{n}\frac{\varphi_n^i}{i!}
=1+\sum_{1\leq i \leq n}\varepsilon_i+\sum_{1\leq i &lt;j\leq
n}\varepsilon_{ij}+\cdots+\varepsilon_{12\cdots n}.
\end{equation} To obtain Eq.~(\ref{exp}) we have used the multinomial theorem and
$\varphi_n^{n+1}=0$ $\forall$ $n\geq 1$. Third, let
$\phi_n\left(\hat{\varepsilon}_i,\hat{\varepsilon}_j, \ldots ,
\hat{\varepsilon}_k\right)$ and $d
\mu_n\left(\hat{\varepsilon}_i,\hat{\varepsilon}_j, \ldots ,
\hat{\varepsilon}_k\right)$ mean $\phi_n$ with
$\varepsilon_i=\varepsilon_j=\cdots=\varepsilon_k=0$ and $d\mu_n$
with $d\varepsilon_i$, $d\varepsilon_j$, $\dots$, $d\varepsilon_k$
omitted, respectively. We have
\begin{equation}\label{IntAux1}
\int\phi_n \varepsilon_{ij \cdots k}d\mu_n=\int\phi_n\left(\hat{\varepsilon}_i,\hat{\varepsilon}_j, \ldots
, \hat{\varepsilon}_k\right) \varepsilon_{ij \cdots k}d \mu_n
=\int\phi_n\left(\hat{\varepsilon}_i,\hat{\varepsilon}_j, \ldots
, \hat{\varepsilon}_k\right) d \mu_n\left(\hat{\varepsilon}_i,\hat{\varepsilon}_j, \ldots
, \hat{\varepsilon}_k\right).
\end{equation} Eq.~(\ref{IntAux1}) follows directly from the general expression in Eq.~(\ref{phi}) and Definition
\ref{Def2}. Finally, from Definition \ref{Def2}, we conclude that
the order of integration is irrelevant, i.e., a Fubini-like
theorem holds in the setting of Grassmann-Berezin integration.

We are now ready to prove Eq.~(\ref{CarBer}).

\section{Proof of Eq.~(\ref{CarBer})}

Let us write $\mathbb{Q}[[z]]$ for the ring of formal power series
in the variable $z$ over $\mathbb{Q}$. We recall the generating
function for the Bernoulli numbers $B_j$ in $\mathbb{Q}[[z]]$
\cite{Wilf}, i.e.,
\begin{equation}\label{GFBer1}
\frac{1}{\sum_{i=0}^{\infty}\frac{z^i}{(i+1)!}}=\frac{z}{e^z-1}=\sum_{j=0}^{\infty}B_j\frac{z^j}{j!}
\end{equation} and, making the change $z \rightarrow -z$ in Eq.~(\ref{GFBer1}), we get
\begin{equation}\label{GFBer2}
\frac{e^z}{\sum_{i=0}^{\infty}\frac{z^i}{(i+1)!}}=\frac{ze^z}{e^z-1}=\sum_{j=0}^{\infty}B_j\frac{(-z)^j}{j!}.
\end{equation}

Following the strategy of our previous work
\cite{NetodAnjos,Neto}, we consider Eqs.~(\ref{GFBer1}) and
(\ref{GFBer2}) in the context of the Zeon algebra with the
replacement $z \rightarrow \phi_k \equiv \varphi_k$. Therefore, we
get
\begin{equation}\label{ZGFBer1}
\frac{1}{\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}}=\sum_{j=0}^{k}B_j\frac{\varphi_k^j}{j!}
\end{equation} and
\begin{equation}\label{ZGFBer2}
\frac{e^{\varphi_k}}{\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}}=\sum_{j=0}^{k}B_j\frac{(-\varphi_k)^j}{j!},
\end{equation} using that $\varphi_k^{k+1}=0$ $\forall$ $k\geq 1$. We observe that
$b(\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!})=1\neq 0$ and, hence,
$\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}$ is invertible in
$\mathcal{Z}_k$.

Now, integrating Eq.~(\ref{ZGFBer1}) in the Zeon algebra and using
Eq.~(\ref{IntAux}) we get
\begin{equation}\label{ZBer1}
\int\frac{1}{\sum_{i=0}^{j}\frac{\varphi_j^i}{(i+1)!}}d\mu_j
=\sum_{k=0}^{j}\frac{B_k}{k!}\int\varphi_j^kd\mu_j=B_j
\end{equation} $\forall$ $j\geq 1$. It is straightforward to verify that
the representation in Eq.~(\ref{ZBer1}) is equivalent to a
well-known representation of the Bernoulli numbers \cite[Theorem
3.1]{Jeong}, i.e.,
$$
B_n=n!\sum_{i=1}^{n}(-1)^i
\sum_{\substack{i_1,i_2, \ldots, i_n \geq 0\\ i_1+i_2+\cdots + i_n=i\\ i_1+2i_2+\cdots + ni_n=n}}
\frac{i!}{i_1!i_2!\cdots i_n!}\frac{1}{2!^{i_1}3!^{i_2}
\cdots (n+1)!^{i_n}}.
$$
Indeed, we have
\begin{align*}
B_n&amp;=\sum_{i=1}^n(-1)^i\int\left(\frac{\varphi_n}{2!}+\frac{\varphi_n^2}{3!}+\cdots
+\frac{\varphi_n^n}{(n+1)!}\right)^id\mu_n\\
&amp;=\sum_{i=1}^n(-1)^i
\sum_{\substack{i_1,i_2, \ldots, i_n \geq 0\\ i_1+i_2+\cdots + i_n=i}}
\frac{i!}{i_1!i_2! \cdots i_n!}\int \frac{\varphi_n^{i_1}\varphi_n^{2i_2}\cdots \varphi_n^{ni_n}}{2!^{i_1}
3!^{i_2}\cdots (n+1)!^{i_n}}d\mu_n\\
&amp;=n!\sum_{i=1}^n(-1)^i\sum_{\substack{i_1,i_2,\ldots, i_n \geq 0\\ i_1+i_2+\cdots + i_n=i}}\frac{i!}{i_1!i_2! \cdots i_n!}
\frac{\delta_{n,i_1+2i_2+\cdots + ni_n}}{2!^{i_1}3!^{i_2} \cdots (n+1)!^{i_n}}\\
&amp;=n!\sum_{i=1}^n(-1)^i\sum_{\substack{i_1,i_2, \ldots, i_n \geq 0\\ i_1+i_2+\cdots + i_n=i}}\frac{i!}{i_1!i_2! \cdots i_n!}
\frac{\delta_{n,i_1+2i_2+\cdots + ni_n}}{2!^{i_1}3!^{i_2} \cdots (n+1)!^{i_n}},
\end{align*} using Eqs.~(\ref{IntAux}), (\ref{Inv}) and the multinomial theorem.

By considering Eq.~(\ref{ZGFBer2}), we take $k=m+n$ and write
$\varphi_{m+n}=\varphi_m+\phi_n$ with
$\varphi_m:=\varepsilon_1+\cdots+\varepsilon_m$,
$\phi_n:=\epsilon_1+\cdots+\epsilon_n$, and
$\epsilon_i:=\varepsilon_{i+m}$ $\forall$ $1\leq i\leq n$. Next,
we multiply both sides of Eq.~(\ref{ZGFBer2}) by $e^{-\phi_n}$.
Finally, integrating the resulting equation with
$d\mu_m:=d\varepsilon_m\cdots d\varepsilon_1$ and
$d\nu_n:=d\epsilon_n\cdots d\epsilon_1$ we get
\begin{equation}\label{ZGFBer3}
\int\left(\int\frac{e^{\varphi_m}}{\sum_{i=0}^{m+n}\frac{(\varphi_{m}+\phi_{n})^i}{(i+1)!}}d\mu_m\right)d\nu_n
=\sum_{j=0}^{m+n}\frac{B_j}{j!}\int\left(\int (-\varphi_m-\phi_n)^je^{-\phi_n}d\nu_n\right)d\mu_m.
\end{equation} In Eq.~(\ref{ZGFBer3}) we have used a Fubini-like argument to perform the integrations. We first consider the left-hand side of Eq.~(\ref{ZGFBer3}).
By expanding $e^{\varphi_m}$ as in Eq.~(\ref{exp}) and integrating
with respect to $d\mu_m$ we will need to analyze terms such as
\begin{eqnarray}\label{ZGFBer4}
&amp;&amp;\sum_{1\leq i_1&lt; i_2 &lt; \cdots &lt;i_j \leq m}\int\left(\int\frac{\varepsilon_{i_1i_2\cdots i_j}}{\sum_{i=0}^{m+n}
\frac{(\varphi_{m}+\phi_{n})^i}{(i+1)!}}d\mu_m\right)d\nu_n\nonumber\\
&amp;&amp;={m \choose j}\int\left(\int\frac{1}{\sum_{i=0}^{m-j+n}\frac{(\varphi_{m-j}+\phi_{n})^i}{(i+1)!}}d\mu_{m-j}\right)d\nu_n
={m \choose j}B_{n+m-j}.
\end{eqnarray}
Therefore, using Eq.~(\ref{ZGFBer4}), we get for the left-hand
side of Eq.~(\ref{ZGFBer3})
\begin{equation}\label{lhs}
\sum_{i=0}^m{m \choose i}B_{n+i}.
\end{equation}
Similarly, we expand $e^{-\phi_n}$ as in Eq.~(\ref{exp}) and
integrate with respect to $d\nu_n$ to obtain for the right-hand
side of Eq.~(\ref{ZGFBer3})
\begin{equation}\label{rhs}
(-1)^{m+n}\sum_{j=0}^n{n \choose j}B_{m+j}.
\end{equation} By equating the expressions in (\ref{lhs}) and (\ref{rhs}) we obtain the desired result, i.e.,
Eq.~(\ref{CarBer}).

Let $B_i^{(j)}$ be the $i$-th Bernoulli number of order $j$ with
generating function in $\mathbb{Q}[[z]]$ given by
$$
\left(\frac{z}{e^z-1}\right)^{j}=\sum_{i=0}^{\infty}B_i^{(j)}\frac{z^i}{i!}.
$$ Note that $B_n^{(1)}\equiv B_n$. Following the procedure just described, it is straightforward to
prove an analogous identity for the Bernoulli numbers of higher
order, i.e.,
$$
\sum_{i=0}^mk^i{m \choose i}B_{n+i}^{(k)}=(-1)^{m+n}\sum_{j=0}^nk^j{n \choose j}B_{m+j}^{(k)}.
$$

\section{Acknowledgments}
The author thanks the anonymous referee for suggestions that
improved the paper.

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\noindent 2010 {\it Mathematics Subject Classification}: Primary
11B68; Secondary 33B10, 05A15, 05A19.

\noindent \emph{Keywords: } Zeon algebra, Berezin integration,
Bernoulli number, generating function.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences \seqnum{A027641} and
\seqnum{A027642}.)

\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received January 29 2015; 
revised version received  April 6 2015.
Published in {\it Journal of Integer Sequences}, May 25 2015.

\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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