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\begin{center}
\vskip 1cm{\LARGE\bf A Note on a Theorem of Schumacher
} \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 307211/2015-0.}\\
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 paper, we give a short new proof of a recent result due to
Schumacher concerning an extension of Faulhaber's identity for the
Bernoulli numbers. Our approach follows from basic manipulations
involving the ordinary generating function for the Bernoulli
polynomials in the context of the Zeon algebra.
\end{abstract}

\section{Introduction}

Zeon algebra \cite[Chap.\ 5]{MansourSchork}, \cite[Chap.\ 2]{SchoSta} and Grassmann algebra \cite[Chap.\ 1]{BerIS}, \cite[Chap.\ 1]{DeWitt}, \cite[Chap.\ 3]{Rogers}, \cite[Chap.\ 2]{SchoSta} are efficient tools towards proving combinatorial identities. In the context of the Zeon algebra, examples include a criterion for ergodicity of Markov chains \cite{Fein}, alternative proofs of Spivey's identity for Bell numbers \cite{NetodAnjos}, the one-variable Fa\`a di Bruno formula \cite{NetodAnjos}, identities involving Stirling numbers of the second kind, Bernoulli numbers, and Bernoulli polynomials \cite{Neto1,Neto2,Neto3}. Building on ideas from Grassmann algebra we mention, e.g., proofs of theorems of the
matrix-tree type \cite{Abde}, representation of the generating function for hyperforests in hypergraphs \cite{Bedi}, Cayley-type identities \cite{Cara}, Lindstr\"om-Gessel-Viennot lemma, and Schur functions \cite{Carro}.

In this paper, we will give yet another example of the utility
of the Zeon algebra by giving a new, simple, and short proof of an extension of
Faulhaber's identity for the Bernoulli numbers \cite[Chap.\ 6]{GKP} obtained recently by Schumacher \cite{Schu}. Another compelling feature of our proof is that it does not assume the usual Faulhaber formula a priori, as in the proof given by Schumacher. More precisely, we will show that
\begin{equation}\label{FauSchu}
\sum_{i=0}^{\lfloor x \rfloor}i^n=\frac{x^{n+1}}{n+1}
+(-1)^n\frac{B_{n+1}}{n+1}+\frac{1}{n+1}\sum_{j=1}^{n+1}(-1)^j{n+1 \choose j}B_j(\{x \})x^{n-j+1},
\end{equation} using the Zeon algebra \cite{MansourSchork,NetodAnjos}. Throughout
this work, we let $\mathbb{N}$, $\mathbb{Q}$, and $\mathbb{R}$ denote the natural, rational, and real numbers, respectively. We define $\mathbb{N}_0:=\{0\}\cup\mathbb{N}$ and $\mathbb{R}_0^+:=\{x \in \mathbb{R}: x \geq 0\}$. For $x \in \mathbb{R}_0^+$, we write $\lfloor x\rfloor$ for the floor
of $x$ and $\{x\}$ for the fractional part of $x$. In Eq.~(\ref{FauSchu}) we take $x \in \mathbb{R}_0^+$ and $n \in \mathbb{N}_0$.

We remark that there are other examples concerning extensions of
Faulhaber's identity in different contexts
\cite{BaMe,BPS,Chap,CFZ,Dube,Eric,GuZe,Hirs,Howa,Kim,Knut,McPa}.

Before we continue, we establish the basic underlying algebraic
setup needed to give the proof of Eq.~(\ref{FauSchu}).

\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
$\{\varepsilon_i\}_{i=1}^n$ ($n<\infty$) 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}

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)
$$\phi_n=
a+a_i\varepsilon_i+a_{ij}\varepsilon_{ij}+\cdots+ a_{12\cdots
n}\varepsilon_{12\cdots n}=\sum_{\mathbf{i} \in
2^{[n]}}a_{\mathbf{i}}\varepsilon_\mathbf{i},$$ 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 $1\leq i<j< \cdots \leq n$. 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 $s(\phi_n)=\phi_n-a$. This terminology is borrowed from
the literature on superanalysis \cite[Chap.\ 1]{DeWitt}.

\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 $d\nu_n:=d\varepsilon_n \cdots d\varepsilon_1$)
$$
d\varepsilon_id\varepsilon_j=d\varepsilon_jd\varepsilon_i,\,\,
\int \phi_n\bigl(\hat{\varepsilon}_i\bigr)d\varepsilon_i=0,\,\,{\rm and}
\int \phi_n\bigl(\hat{\varepsilon}_i\bigr)\varepsilon_id\varepsilon_i=\phi_n\bigl(\hat{\varepsilon}_i\bigr),
$$ 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.,
$$
\int f(\phi_n) d\nu_n = \int \cdots \biggl(\int \biggl(\int f(\phi_n) d\varepsilon_n\biggr)
d \varepsilon_{n-1}\biggr)\cdots d\varepsilon_1.
$$\end{definition}

We now extend some of the constructions of previous work on Zeons
and Bernoulli numbers \cite{Neto2} to the context of Bernoulli
polynomials \cite[Chap.\ 7]{GKP}, \cite[Chap.\ 4]{Wilf}. Let us
write $\mathbb{Q}[[x,z]]$ for the ring of formal power series in
the variables $x$ and $z$ over $\mathbb{Q}$. We recall the
generating function for the Bernoulli polynomials $B_j(x)$ in
$\mathbb{Q}[[x,z]]$, i.e.,
\begin{equation}\label{GFBerP1}
\frac{e^{xz}}{\sum_{i=0}^{\infty}\frac{z^i}{(i+1)!}}=\frac{ze^{xz}}{e^z-1}=\sum_{j=0}^{\infty}B_j(x)\frac{z^j}{j!}
\end{equation} and, making the change $z \mapsto -z$ in Eq.~(\ref{GFBerP1}), we get
\begin{equation}\label{GFBerP2}
\frac{e^{\left(1-x\right)z}}{\sum_{i=0}^{\infty}\frac{z^i}{(i+1)!}}=\frac{ze^{\left(1-x\right)z}}{e^z-1}=\sum_{j=0}^{\infty}B_j(x)\frac{(-z)^j}{j!}.
\end{equation} Note that $B_j(0)\equiv B_j$ are the Bernoulli numbers.

Following the strategy of our previous work
\cite{NetodAnjos,Neto2}, we consider Eqs.~(\ref{GFBerP1}) and
(\ref{GFBerP2}) in the context of the Zeon algebra with the
replacement $z \rightarrow \phi_k \equiv \varphi_k:=\varepsilon_1+\cdots+\varepsilon_k$. Therefore, we
get
\begin{equation}\label{ZGFBerP1}
\frac{e^{x\varphi_k}}{\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}}=\sum_{j=0}^{k}B_j(x)\frac{\varphi_k^j}{j!}
\end{equation} and
\begin{equation}\label{ZGFBerP2}
\frac{e^{\left(1-x\right)\varphi_k}}{\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}}=\sum_{j=0}^{k}B_j(x)\frac{(-\varphi_k)^j}{j!},
\end{equation} using that $\varphi_k^{k+1}=0$ $\forall$ $k\geq 1$. We observe that
\begin{equation}\label{b1}b\left(\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}\right)=1\neq 0\end{equation} and, hence,
$\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}$ is invertible in
$\mathcal{Z}_k$.

Now, integrating Eq.~(\ref{ZGFBerP1}) in the Zeon algebra and using Definition \ref{Def2}, we get the representation of the Bernoulli polynomials
$$
\int\frac{e^{x\varphi_k}}{\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}}d\nu_k
=\sum_{j=0}^{k}\frac{B_j(x)}{j!}\int\varphi_k^jd\nu_k=B_k(x)$$ $\forall$ $k\geq 1$.

Similarly, integrating Eq.~(\ref{ZGFBerP2}) in the Zeon algebra, we get
\begin{equation}\label{ZBerP}
\int\frac{e^{\left(1-x\right)\varphi_k}}{\sum_{i=0}^{k}\frac{\varphi_k^i}{(i+1)!}}d\nu_k
=\sum_{j=0}^{k}(-1)^j\frac{B_j(x)}{j!}\int\varphi_k^jd\nu_k=(-1)^kB_k(x)
\end{equation} $\forall$ $k\geq 1$.

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

We are now ready to prove Eq.~(\ref{FauSchu}). We start with the following identity
$$
e^{\left(\lfloor x \rfloor+1\right)\varphi_{n+1}}-e^{\varphi_{n+1}}=\sum_{i=1}^{\lfloor x \rfloor}e^{i\varphi_{n+1}}\left(e^{\varphi_{n+1}}-1\right)=\varphi_{n+1}\sum_{i=1}^{\lfloor x \rfloor}e^{i\varphi_{n+1}}\left(\sum_{j=0}^{n+1}\frac{\varphi_{n+1}^j}{(j+1)!}\right).
$$ Note that Eq.~(\ref{b1}) can be used and we can write
$$
\frac{e^{\left(\lfloor x \rfloor+1\right)\varphi_{n+1}}-e^{\varphi_{n+1}}}{\sum_{j=0}^{n+1}\frac{\varphi_{n+1}^j}{(j+1)!}}=\varphi_{n+1}\sum_{i=1}^{\lfloor x \rfloor}e^{i\varphi_{n+1}}.$$

Next, using the Grassmann-Berezin integration of Definition \ref{Def2}, we have
\begin{equation}\label{ZSchu}
\int \frac{e^{\left(\lfloor x \rfloor+1\right)\varphi_{n+1}}-e^{\varphi_{n+1}}}{\sum_{j=0}^{n+1}\frac{\varphi_{n+1}^j}{(j+1)!}}d\nu_{n+1}=\int \varphi_{n+1}\sum_{i=1}^{\lfloor x \rfloor}e^{i\varphi_{n+1}}d\nu_{n+1}.
\end{equation} We observe that
\begin{equation}\label{SchuP1}
\int \varphi_{n+1}\sum_{i=1}^{\lfloor x \rfloor}e^{i\varphi_{n+1}}d\nu_{n+1}
=(n+1)\sum_{i=1}^{\lfloor x \rfloor}\int e^{i\varphi_n}d\nu_n
=(n+1)\sum_{i=1}^{\lfloor x \rfloor}i^n.
\end{equation} Using  Eq.~(\ref{ZBerP}) and $x=\lfloor x\rfloor + \{x \}$, we obtain
\begin{eqnarray}\label{SchuP2}
\int \frac{e^{\left(\lfloor x \rfloor+1\right)\varphi_{n+1}}}{\sum_{j=0}^{n+1}\frac{\varphi_{n+1}^j}{(j+1)!}}d\nu_{n+1}&=&
\int \frac{e^{\left(x-\{ x \}+1\right)\varphi_{n+1}}}{\sum_{j=0}^{n+1}\frac{\varphi_{n+1}^j}{(j+1)!}}d\nu_{n+1}\nonumber\\
&=&
\sum_{k=0}^{n+1}\frac{x^k}{k!}\int \frac{e^{\left(1-\{ x \}\right)\varphi_{n+1}}}{\sum_{j=0}^{n+1}\frac{\varphi_{n+1}^j}{(j+1)!}}\varphi_{n+1}^kd\nu_{n+1}\nonumber\\
&=&x^{n+1}+\sum_{k=0}^n{n+1 \choose k}x^k\int \frac{e^{\left(1-\{ x \}\right)\varphi_{n-k+1}}}{\sum_{j=0}^{n-k+1}\frac{\varphi_{n-k+1}^j}{(j+1)!}}d\nu_{n-k+1}\nonumber\\
&=&x^{n+1}+\sum_{k=0}^n{n+1 \choose k}(-1)^{n-k+1}B_{n-k+1}(\{x \})x^k\nonumber\\
&=&x^{n+1}+\sum_{k=1}^{n+1}{n+1 \choose k}(-1)^kB_{k}(\{x \})x^{n-k+1}
\end{eqnarray} making the change of variables $n-k+1 \mapsto k$ to obtain the last equality.
Finally, taking $x=0$ in Eq.~(\ref{ZBerP}), we get
\begin{equation}\label{SchuP3}
\int \frac{e^{\varphi_{n+1}}}{\sum_{j=0}^{n+1}\frac{\varphi_{n+1}^j}{(j+1)!}}d\nu_{n+1}=(-1)^{n+1}B_{n+1}.
\end{equation}
Collecting the results in Eqs.~(\ref{SchuP1}), (\ref{SchuP2}), (\ref{SchuP3}), and going back to Eq.~(\ref{ZSchu}), we arrive at the desired result, i.e., Eq.~(\ref{FauSchu}).

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

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\end{thebibliography}

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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, Faulhaber's formula.

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

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\vspace*{+.1in}
\noindent
Received June 19 2016;
revised version received October 1 2016.
Published in {\it Journal of Integer Sequences}, October 10 2016.

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