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\begin{center}
\vskip 1cm{\LARGE\bf The Dual of Spivey's Bell Number Identity  \\
\vskip .1in
from 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 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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\newcommand {\stirlingf}[2]{\genfrac[]{0pt}{}{#1}{#2}}

\begin{abstract}
In this paper, we give a new short proof of the dual of Spivey's Bell number identity due to Mez\H{o}. Our approach follows from basic manipulations involving a fundamental identity representing factorials in the Zeon algebra. This work, along with a previous one due to the author and dos Anjos, shows that Spivey's and Mez\H{o}'s identities have at their root a common underlying algebraic origin.
\end{abstract}

\section{Introduction}

In this paper, we will give a new, simple, and short proof of the dual of Spivey's Bell number identity obtained by Mez\H{o} \cite[Chap.\ 3]{MansourSchork}, \cite{Mezo}. We mention that the original proofs of the identities due to Spivey and Mez\H{o} are combinatorial in nature \cite{GouQua,Mezo,Spi}. Indeed, the proof given by Mez\H{o} is constructed by considering the enumeration of the permutations of $m+n$ elements in terms of the number of $k$-permutations of $n$ and the number of permutations of $m$ with $j$ disjoint cycles. This work, along with a previous work concerning the proof of Spivey's identity from Zeons \cite{NetodAnjos}, adds another point of view in the origin of Spivey's and Mez\H{o}'s results which is algebraic in nature, as a direct consequence of the use of the Zeon algebra \cite[Chap.\ 5]{MansourSchork}, \cite{NetodAnjos}, \cite[Chap.\ 2]{SchoSta}. Throughout this
work we let $\mathbb{R}$ denote the real numbers and $\mathbb{N}$ the
positive integers.

More precisely, if $m$, $n$ $\in$ $\mathbb{N}$,
we will show that
\begin{equation}\label{Mezo}
(m+n)!= \sum_{k=0}^n\sum_{j=0}^m m^{\overline{n-k}} \stirlingf
{m}{j}{n \choose k}k!
\end{equation} using the Zeon algebra. In Eq.~(\ref{Mezo}) $a^{\overline{n}}=a(a+1)\cdots(a+n-1)$ is the Pochhammer symbol with $a \in \mathbb{R}$, ${n \choose k}=n!/(k!(n-k)!)$ is the
binomial coefficient, and $\stirlingf
{m}{j}$ are the unsigned Stirling numbers of the first kind \cite[Chap.\ 6]{GKP}, \cite[Eq.\ (3.5.3)]{Wilf}.

For completeness, in the next section, we introduce the tools needed to give the proof of Eq.~(\ref{Mezo}). More precisely, we give the definition of the Zeon algebra and the Grassmann-Berezin integral
in the Zeon algebra. 

\section{Brief review of the Zeon algebra and Grassmann-Berezin integral
in the Zeon algebra}

\begin{definition}\label{Def1} The
\textit{Zeon algebra} $\mathcal{Z}_n \supset \mathbb{R}$ is
defined as the associative and commutative algebra generated by the collection
$\{\varepsilon_i\}_{i=1}^n$ ($n<\infty$) of nilpotent elements and the scalar $1 \in
\mathbb{R}$, which is the identity of the algebra.
\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 with the convention of sum over
repeated indices implicit and taking $\varepsilon_{\emptyset}=1$ as
$$\vartheta_n=
a+a_i\varepsilon_i+a_{ij}\varepsilon_{ij}+\cdots+ a_{12\cdots
n}\varepsilon_{12\cdots n}=\sum_{\mathbf{i} \in
2^{[n]}}a_{\mathbf{i}}\varepsilon_\mathbf{i},$$ 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 $\vartheta_n$ and write $b(\vartheta_n):=a$ and to
$\vartheta_n-a$ as the soul such that $s(\vartheta_n):=\vartheta_n-a$. Note that
$s^{n+1}(\vartheta_n)=0$.

As described in previous work \cite{NetodAnjos} a real analytic function $f$ can be extended to the realm of the Zeon algebra taking 
\begin{equation}\label{f}
f(\vartheta_n):=\sum_{i=0}^n\frac{f^{(i)}\bigl(b\bigl(\vartheta_n\bigr)\bigr)}{i!}s^i\bigl(\vartheta_n\bigr)
=\sum_{i=0}^n\frac{f^{(i)}\bigl(a\bigr)}{i!}s^i\bigl(\vartheta_n\bigr),
\end{equation} where $f^{(i)}(a)=d^if(x)/dx^i|_{x=a}$ is the $i$-th ordinary derivative of $f(x)$ at $a$.
Note that $f\bigl(a+s\bigl(\vartheta_n\bigr)\bigr)|_{s(\vartheta_n)=0}=f(a)$
and $f\bigl(a+s\bigl(\vartheta_n\bigr)\bigr) \in \mathcal{Z}_n$,
because $s^{n+1}(\vartheta_n)=0$.

We consider specific examples of Eq.~(\ref{f}) important for this work. If
$a\neq 0$, then $\vartheta_n$ is invertible and the inverse is given
by  
\begin{equation}\begin{array}{lll}\label{InvZ}
\vartheta_n^{-1}
:=\displaystyle\frac{1}{a}
\left(1-\displaystyle\frac{s(\vartheta_n)}{a}
+\displaystyle\frac{s^2(\vartheta_n)}{a^2}
+\cdots+(-1)^n\displaystyle\frac{s^n(\vartheta_n)}{a^n}\right).
\end{array}\end{equation} 

Next, we will need 
\begin{equation}\label{ExpZ}
e^{\vartheta_n}:=e^a\sum_{m=0}^n\frac{s^{m}\left(\vartheta_n\right)}{m!}.
\end{equation}

As a final example, for $\vartheta_n=1+s(\vartheta_n)$, we consider
\begin{equation}\label{LnZ}
\ln\left(1+s\left(\vartheta_n\right)\right):=\sum_{m=1}^{n}(-1)^{m+1}\frac{s^m(\vartheta_n)}{m},
\end{equation} where the right-hand side of Eq.~(\ref{LnZ}) is the solution of the equation
$$ e^{\ln\left(1+s(\vartheta_n)\right)}\equiv 1+s(\vartheta_n).
$$

We give some examples of the functions considered in Eqs.~(\ref{InvZ}), (\ref{ExpZ}), and (\ref{LnZ}) for $n=3$. We have
$$
\frac{1}{1+\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-5\varepsilon_{123}}
=1-\frac{1}{2}\varepsilon_2-\sqrt{3}\varepsilon_{13}+\left(5+\sqrt{3}\right)\varepsilon_{123},
$$
$$
e^{1+\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-5\varepsilon_{123}}
=e\left(1+\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}+\left(\frac{\sqrt{3}}{2}-5\right)\varepsilon_{123}\right),
$$ and
$$
\ln \left(1+\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-5\varepsilon_{123}\right)
=\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-\left(5+\frac{\sqrt{3}}{2}\right)\varepsilon_{123}.
$$
Note that $e^{\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-\left(5+\frac{\sqrt{3}}{2}\right)\varepsilon_{123}}=1+\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-5\varepsilon_{123}$.


\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 
$$
d\varepsilon_id\varepsilon_j=d\varepsilon_jd\varepsilon_i,\,\,
\int \vartheta_n\bigl(\hat{\varepsilon}_i\bigr)d\varepsilon_i=0,\,\,{\rm and}
\int \vartheta_n\bigl(\hat{\varepsilon}_i\bigr)\varepsilon_id\varepsilon_i=\vartheta_n\bigl(\hat{\varepsilon}_i\bigr),
$$ where $\vartheta_n\bigl(\hat{\varepsilon}_i\bigr)$ means any element of $\mathcal{Z}_n$ with no dependence on
$\varepsilon_i$. We use throughout this work the
compact notation $d\nu_n:=d\varepsilon_n \cdots d\varepsilon_1$. Multiple integrals are iterated integrals, i.e.,
$$
\int f(\vartheta_n) d\nu_n = \int \cdots \biggl(\int \biggl(\int f(\vartheta_n) d\varepsilon_n\biggr)
d \varepsilon_{n-1}\biggr)\cdots d\varepsilon_1.
$$\end{definition} The standard literature on Grassmann algebra comprises, e.g., the work of Berezin \cite[Chap.\ 1]{BerIS}, DeWitt \cite[Chap.\ 1]{DeWitt}, and Rogers \cite[Chap.\ 3]{Rogers}.

Some examples of the integration in Definition \ref{Def2} are given below. We have

$$
\int \frac{1}{1+\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-5\varepsilon_{123}}
d\varepsilon_1=-\sqrt{3}\varepsilon_3+\left(5+\sqrt{3}\right)\varepsilon_{23}
$$
and
$$
\int e^{1+\frac{1}{2}\varepsilon_2+\sqrt{3}\varepsilon_{13}-5\varepsilon_{123}}
d\varepsilon_1d\varepsilon_3=e\left(\sqrt{3}+\left(\frac{\sqrt{3}}{2}-5\right)\varepsilon_{2}\right).$$

A particularly simple result which follows from the multinomial theorem is
\begin{equation}\label{FunId}
\int \varphi_n^nd\nu_n=n!
\end{equation} with $\vartheta_n=\varphi_n:=\varepsilon_1+\cdots+\varepsilon_n$.

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

We are now ready to prove Eq.~(\ref{Mezo}). Let us take $\phi_m=\epsilon_1+\cdots + \epsilon_m$ with $\epsilon_i:=\varepsilon_{n+i}$, $i=1,\ldots,m$, such that $\varphi_{n+m}=\phi_m+\varphi_n \in \mathcal{Z}_{m+n}$.
 We start with the following identity
\begin{equation}\label{FacInvZmn}
(m+n)!=\int (\phi_m+\varphi_n)^{m+n}d\mu_md\nu_n=\int \sum_{k=0}^{m+n}
(\phi_m+\varphi_n)^kd\mu_md\nu_n=\int \frac{1}{1-\phi_m-\varphi_n}d\mu_md\nu_n,
\end{equation} using Eqs.~(\ref{InvZ}) and (\ref{FunId}). Next, we observe that 
$$\begin{array}{lcl}
(m+n)!
&=&\displaystyle\int \frac{1}{1-\varphi_n}\frac{1}{1-\frac{\phi_m}{1-\varphi_n}}d\mu_md\nu_n\\
&=& \displaystyle\sum_{k=0}^n\int \varphi_n^k\frac{1}{1-\frac{\phi_m}{1-\varphi_n}}d\mu_md\nu_n\\
&=&\displaystyle\int \frac{1}{1-\frac{\phi_m}{1-\varphi_{n}}}d\mu_m d\nu_{n}+ \sum_{k=1}^n\sum_{1 \leq j_1 < \cdots < j_k \leq n}k!\int \varepsilon_{j_1\cdots j_k} \frac{1}{1-\frac{\phi_m}{1-\varphi_{n}}}d\mu_m d\nu_{n}\\
&=& \displaystyle\sum_{k=0}^n{n \choose k}k!\int \frac{1}{1-\frac{\phi_m}{1-\varphi_{n-k}}}d\mu_m d\nu_{n-k}\\
&=& \displaystyle\sum_{k=0}^n{n \choose k}k!\int \frac{1-\varphi_{n-k}}{1-\phi_m-\varphi_{n-k}}d\mu_m d\nu_{n-k}\\
&=& \displaystyle\sum_{k=0}^n{n \choose k}k!\left(\int \frac{1}{1-\phi_m-\varphi_{n-k}}d\mu_m d\nu_{n-k}-\int \frac{n-k}{1-\phi_m-\varphi_{n-k-1}}d\mu_m d\nu_{n-k-1}\right)\\
&=& \displaystyle\sum_{k=0}^n{n \choose k}k!\left(\left(m+n-k\right)!-\left(n-k\right)\left(m+n-k-1\right)!\right)\\
&=&\displaystyle\sum_{k=0}^n{n \choose k}k!m\left(m+n-k-1\right)!,
\end{array}$$ using Eq.~(\ref{FacInvZmn}). Finally, recalling the definition of $m^{\overline{n-k}}$ in Eq.~(\ref{Mezo}), we obtain
\begin{equation}\label{PreMezo}
(m+n)!= m!\sum_{k=0}^n{n \choose k}k!m^{\overline{n-k}}.
\end{equation}

Now, we consider the unsigned Stirling numbers of the first kind $\stirlingf{n}{j}$ given by the generating function \cite[Eq.\ (3.5.3)]{Wilf}
\begin{equation}\label{SG}
\frac{1}{j!}\left(\ln\frac{1}{1-x}\right)^j=\sum_{n=j}^{\infty}\stirlingf{n}{j}\frac{x^n}{n!}.
\end{equation} Using Eqs.~(\ref{LnZ}) and (\ref{FunId}) we arrive at
$$
\stirlingf
{m}{j}=\frac{1}{j!}\int \left(\ln\frac{1}{1-\phi_m}\right)^jd\mu_m,
$$ extending Eq.~(\ref{SG}) to the context of Zeons by following previous work \cite{NetodAnjos,Neto}. Note that
\begin{equation}\label{S1stFac}
\sum_{j=0}^m\stirlingf
{m}{j}
=\int e^{\ln \left(\frac{1}{1-\phi_m}\right)}d\mu_m
=\int \frac{1}{1-\phi_m}d\mu_m=m!,
\end{equation} using Eqs.~(\ref{InvZ}), (\ref{ExpZ}), (\ref{LnZ}), and (\ref{FunId}).

Finally, from Eqs.~(\ref{PreMezo}) and (\ref{S1stFac}), we arrive at Eq.~(\ref{Mezo}).

\begin{thebibliography}{99}

\bibitem{BerIS} F. A. Berezin, {\it Introduction to Superanalysis}, Reidel Publishing Company, 1987.

\bibitem{DeWitt} B. DeWitt, {\it Supermanifolds}, Cambridge University Press, 1992.

\bibitem{GouQua}
H. W. Gould and J. Quaintance, Implications of Spivey's Bell number
formula, {\it J. Integer Seq.} \textbf{11} (2008), 
\href{http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Gould/gould35.html}{Article 08.3.7}.

\bibitem{GKP} R. L. Graham, D. E. Knuth, and O. Patashnik, {\it Concrete Mathematics}, Addison-Wesley, second edition, 1994.

\bibitem{MansourSchork} T. Mansour and M. Schork, {\it Commutation
Relations, Normal Ordering, and Stirling Numbers}, Chapman and Hall/CRC
Press, 2015.

\bibitem{Mezo} I. Mez\H{o}, The dual of Spivey's Bell number formula, {\it J. Integer Seq.} {\bf 15} (2012),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL15/Mezo/mezo14.html}{Article 12.2.4}.

\bibitem{NetodAnjos}
A. F. Neto and P. H. R. dos Anjos, Zeon algebra and combinatorial
identities, {\it SIAM Rev.} {\bf 56} (2014), 353--370.

\bibitem{Neto} A. F. Neto, A note on a theorem of Guo, Mez\H{o}, and Qi, {\it J. Integer Seq.} {\bf 19} (2016),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL19/Neto/neto10.html}{Article 16.4.8}.

\bibitem{Rogers} A. Rogers, {\it Supermanifolds: Theory and Applications}, World
Scientific Publishing, 2007.

\bibitem{SchoSta} R. Schott and G. S. Staples, {\it Operator Calculus on Graphs}, Imperial College Press,
2012.

\bibitem{Spi}
M. Z. Spivey, A generalized recurrence for Bell numbers, {\it J.
Integer Seq.} \textbf{11} (2008), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Spivey/spivey25.html}{Article 08.2.5}.

\bibitem{Wilf} H. S. Wilf, {\em Generatingfunctionology},
Academic Press, New York, 1990. Available at
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\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}: Primary
11B73; Secondary 33B10, 05A15, 05A19.

\noindent \emph{Keywords: } Zeon algebra, Berezin integration,
Stirling number of the first kind, Pochhammer symbol, binomial coefficient, generating function.

\bigskip
\hrule
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\noindent (Concerned with sequences \seqnum{A000142} and
\seqnum{A132393}.)

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\vspace*{+.1in}
\noindent
Received August 1 2016;
revised version received November 26 2016.
Published in {\it Journal of Integer Sequences}, December 27 2016.

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