\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem {Congruence}{Congruence} \begin{center} \vskip 1cm{\LARGE\bf A Note on a Theorem of Guo, Mez\H{o}, and Qi} \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}\\ \end{center} \vskip .2 in \begin{abstract} In a recent paper, Guo, Mez\H{o}, and Qi proved an identity representing the Bernoulli polynomials at non-negative integer points $m$ in terms of the $m$-Stirling numbers of the second kind. In this note, using a new representation of the Bernoulli polynomials in the context of the Zeon algebra, we give an alternative proof of the aforementioned identity. \end{abstract} \section{Introduction} In an interesting recent paper \cite{GMQ}, Guo, Mez\H{o}, and Qi found the following identity \begin{equation}\label{Bm-S2nd} B_n(m)=\sum_{l=0}^n(-1)^l\frac{l!}{l+1}S_m(n+m,l+m) \end{equation} relating the Bernoulli polynomials $B_n(m)$ at non-negative integer points $m$ with $m$-Stirling numbers of the second kind $S_m(n+m,l+m)$. Eq.~(\ref{Bm-S2nd}) is a generalization of the identity \cite[p.\ 560]{GKP} $$ B_n=\sum_{m=0}^n(-1)^m\frac{m!}{m+1}S(n,m). $$ Note that $B_n\equiv B_n(0)$ and $S\equiv S_0$ are the usual Bernoulli numbers \cite[Chap.\ 2]{Wilf} and Stirling numbers of the second kind \cite[Chap.\ 1]{Wilf}, respectively. In this work, we will give another proof of (\ref{Bm-S2nd}) by showing that (\ref{Bm-S2nd}) is a straightforward consequence of a new Zeon representation \cite{Fein,NetodAnjos}, \cite[Chap.\ 5]{MansourSchork} of the Bernoulli polynomials. We believe the approach here is of interest because it gives a straightforward demonstration of Guo, Mez\H{o}, and Qi result, and, as a consequence, it provides another instance where computations involving Zeons and/or Grassmann variables provide direct and interesting results \cite{Abde, Bedi,Cara,NetodAnjos,Neto1,Neto2,Scho}. For more on Grassmann variables we refer the reader to the books of Berezin \cite[Chap.\ 1]{BerIS}, DeWitt \cite[Chap.\ 1]{DeWitt}, and Rogers \cite[Chap.\ 3]{Rogers}. For completeness, we recall some basic definitions and results already stated in previous work \cite{NetodAnjos}. Throughout this work we let $\mathbb{R}$ denote the real numbers, $\mathbb{N}$ the positive integers, and $\mathbb{N}_0=\{0\}\cup \mathbb{N}$ the non-negative integers. \section{Zeon Algebra and 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) \begin{equation}\label{phin}\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 $1\leq i n$. Likewise, we have \begin{equation}\label{definv} \frac{1}{\phi_n}=\frac{1}{a+s\left(\phi_n\right)}:=\frac{1}{a}\sum_{m=0}^n\left(-\frac{1}{a}\right)^ms^m\left(\phi_n\right). \end{equation} \section{Proof of Eq.~(\ref{Bm-S2nd})} We are now ready to prove Eq.~(\ref{Bm-S2nd}). We take $\varphi_n:= \varepsilon_1+\cdots + \varepsilon_n \in \mathcal{Z}_n$ from now on. We start with \begin{equation}\label{ZBm-S2nd} B_n(x)=\sum_{m=0}^n \frac{(-1)^m}{m+1}\int e^{x\varphi_n}\left(e^{\varphi_n}-1\right)^m d\nu_n. \end{equation} We will proceed by showing that \begin{equation}\label{Bn} \sum_{m=0}^n{n+1 \choose m}B_m=0 \end{equation} with $n \in \mathbb{N}$, $B_0\equiv 1$, ${n \choose m}:=n!/\bigl(m!(n-m)!\bigr)$, and \begin{equation}\label{Bp} B_n(x)=\sum_{m=0}^n{n \choose m}B_mx^{n-m}. \end{equation} Eqs.~(\ref{Bn}) and (\ref{Bp}) can be regarded as the definitions of the Bernoulli numbers \cite[Eq.\ (6.79)]{GKP} and the Bernoulli polynomials \cite[Eq.\ (7.80)]{GKP}, respectively. We will first show by induction on $n$ that \begin{equation}\label{varphiind} \varphi_n=\displaystyle\sum_{m=0}^{n-1}\frac{\left(-1\right)^m}{m+1}\left(e^{\varphi_n}-1\right)^{m+1}. \end{equation} Indeed, it is easy to see that both sides of Eq.~(\ref{varphiind}) give $\varphi_1\equiv \varepsilon_1$ for $n=1$. Next, using Eqs.~(\ref{defe}) and (\ref{definv}), we have $$\begin{array}{lcl} \varphi_{n+1}&=&\varphi_n+\varepsilon_{n+1}\\ &=&\varphi_n+\varepsilon_{n+1}\displaystyle\frac{e^{\varphi_n}}{1+\left(e^{\varphi_n}-1\right)}\\ &=&\varphi_n+\varepsilon_{n+1}\displaystyle\sum_{m=0}^n(-1)^me^{\varphi_n}\left(e^{\varphi_n}-1\right)^m\\ &=&\displaystyle\sum_{m=0}^{n-1}\frac{\left(-1\right)^m}{m+1}\left(e^{\varphi_n}-1\right)^{m+1}+\varepsilon_{n+1}\sum_{m=0}^n(-1)^me^{\varphi_n}\left(e^{\varphi_n}-1\right)^m\\ &=&\displaystyle\sum_{m=0}^{n-1}\frac{\left(-1\right)^m}{m+1}{m+1 \choose 0}\left(e^{\varphi_n}-1\right)^{m+1}+\varepsilon_{n+1}\sum_{m=0}^n\frac{(-1)^m}{m+1}{m+1 \choose 1} e^{\varphi_n}\left(e^{\varphi_n}-1\right)^m\\ &=&\displaystyle\sum_{m=0}^n\frac{(-1)^m}{m+1} \left(e^{\varphi_n}+\varepsilon_{n+1}e^{\varphi_n}-1\right)^{m+1}\\ &=&\displaystyle\sum_{m=0}^n\frac{(-1)^m}{m+1}\left(e^{\varphi_{n+1}}-1\right)^{m+1}, \end{array}$$ and the result follows, i.e., Eq.~(\ref{varphiind}) is true for all $n\geq1$. Now we can prove Eq.~(\ref{Bn}). Starting with Eq.~(\ref{varphiind}) and using Eq.~(\ref{defe}) we have $$ \varphi_n=\displaystyle\sum_{m=0}^{n-1}\frac{\left(-1\right)^m}{m+1}\sum_{l=1}^n\frac{\varphi_n^l}{l!}\left(e^{\varphi_n}-1\right)^m. $$ Integrating, we get ($n \geq 2$) $$\begin{array}{lcl} 0=\displaystyle\int\varphi_nd\nu_n &=&\displaystyle\sum_{l=1}^n\sum_{1\leq k_1,k_2,\ldots,k_l\leq n}\displaystyle\sum_{m=0}^{n-1}\frac{\left(-1\right)^m}{m+1}\int\varepsilon_{k_1k_2\cdots k_l}\left(e^{\varphi_n}-1\right)^md\nu_n\\ &=&\displaystyle\sum_{l=1}^n{n \choose l}\displaystyle\sum_{m=0}^{n-l}\frac{\left(-1\right)^m}{m+1}\int\left(e^{\varphi_{n-l}}-1\right)^md\nu_{n-l}\\ &=&\displaystyle\sum_{l=1}^n{n \choose l}B_{n-l} \end{array}$$ and making the change of variables $n-l \mapsto l$ we obtain Eq.~(\ref{Bn}). We will now show Eq.~(\ref{Bp}). Indeed, from Eq.~(\ref{ZBm-S2nd}) we have $$\begin{array}{lcl} B_n(x)&=&\displaystyle\sum_{l=0}^n\displaystyle\sum_{m=0}^n \frac{(-1)^l}{l+1}\frac{x^m}{m!}\int \varphi_n^m\left(e^{\varphi_n}-1\right)^l d\nu_n\\ &=&\displaystyle\sum_{l=0}^n \frac{(-1)^l}{l+1}\int \left(e^{\varphi_n}-1\right)^l d\nu_n\\ &&\hspace{2cm}+\displaystyle\sum_{l=0}^n\sum_{m=1}^n\displaystyle\sum_{1\leq l_1,l_2,\ldots,l_m\leq n} \frac{(-1)^l}{l+1}x^m\int \varepsilon_{l_1l_2\cdots l_m}\left(e^{\varphi_n}-1\right)^l d\nu_n\\ &=&\displaystyle\sum_{m=0}^n{n \choose m}x^m\displaystyle\sum_{l=0}^{n-m}\frac{(-1)^l}{l+1}\int \left(e^{\varphi_{n-m}}-1\right)^l d\nu_{n-m}\\ &=&\displaystyle\sum_{m=0}^n{n \choose m}x^mB_{n-m} \end{array}$$ and making the change of variables $n-m \mapsto m$ we obtain Eq.~(\ref{Bp}). We recall the generating function for the $m$-Stirling numbers \cite[Thm.\ 16]{Broder} \begin{equation}\label{GFm-S2nd} \sum_{n=l}^{\infty}S_m(n+m,l+m)\frac{x^n}{n!}=\frac{1}{l!}e^{mx}\left(e^x-1\right)^l \end{equation} with $m \in \mathbb{N}_0$. The Zeon representation of $S_m(n+m,l+m)$ comes from the generating function in Eq.~(\ref{GFm-S2nd}) taking, as in previous work \cite{NetodAnjos}, $x \rightarrow \varphi_n \in \mathcal{Z}_n$ and doing a Grassmann-Berezin integration over the Zeon algebra to get the representation \begin{equation}\label{Zm-S2nd} S_m(n+m,l+m)=\sum_{k=l}^{n}S_m(k+m,l+m)\underbrace{\int \frac{\varphi_n^k}{k!}d\nu_n}_{\delta_{k,n}} =\frac{1}{l!}\int e^{m\varphi_n}\left(e^{\varphi_n}-1\right)^ld\nu_n \end{equation} with $\delta_{k,n}$ meaning the Kronecker delta. We note that the representation in Eq.~(\ref{Zm-S2nd}) is a generalization of the representation of the usual Stirling numbers of the second kind \cite[Prop.\ 2.1]{Scho} obtained by setting $m=0$ in Eq.~(\ref{Zm-S2nd}). Therefore, by setting $x\equiv m$ in Eq.~(\ref{ZBm-S2nd}), we conclude that $$ B_n(m)=\sum_{l=0}^n \frac{(-1)^l}{l+1}\int e^{m\varphi_n}\left(e^{\varphi_n}-1\right)^l d\nu_n, $$ which is equivalent to Eq.~(\ref{Bm-S2nd}) using the Zeon representation of the numbers $S_m(n+m,l+m)$ in Eq.~(\ref{Zm-S2nd}). \section{Acknowledgments} The author thanks the anonymous referee for suggestions that improved the paper. \begin{thebibliography}{99} \bibitem{Abde} A. Abdesselam, The Grassmann-Berezin calculus and theorems of the matrix-tree type, {\it Adv. Appl. Math.} {\bf 33} (2004), 51--70. \bibitem{Bedi} A. Bedini, S. Caracciolo, and A. Sportiello, Hyperforests on the complete hypergraph by Grassmann integral representation, {\it J. Phys. 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Staples, Partitions and Clifford algebras, {\it European J. Combin.} {\bf 29} (2008), 1133--1138. \bibitem{Wilf} H. S. Wilf, {\em Generatingfunctionology}, Academic Press, New York, 1990. Available at \url{http://www.math.upenn.edu/~wilf/DownldGF.html}. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11B68; Secondary 11B73; 33B10; 05A15; 05A19. \noindent \emph{Keywords: } Zeon algebra, Berezin integration, Bernoulli number, $m$-Stirling number, generating function. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A027641}, \seqnum{A027642}, \seqnum{A143494}, \seqnum{A143495}, and \seqnum{A143496}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received March 13 2016; revised version received April 13 2016. Published in {\it Journal of Integer Sequences}, May 11 2016. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}