% EJC papers *must* begin with the following two lines. \documentclass[12pt]{article} \usepackage{e-jc} \specs{P2.27}{21(2)}{2014} % Please remove all other commands that change parameters such as % margins or pagesizes. % only use standard LaTeX packages % only include packages that you actually need % we recommend these ams packages \usepackage{amsthm,amsmath,amssymb} % we recommend the graphicx package for importing figures \usepackage{graphicx} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} % all overfull boxes must be fixed; % i.e. there must be no text protruding into the margins % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} \allowdisplaybreaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \title{{\bf A combinatorial identity of multiple\\ zeta values with even arguments}\thanks{This work was supported by Natural Science Foundation of China (Nos.11101245, 61271355, 11271208, 61202362, 11301302), China Postdoctoral Science Foundation (No.2013M530869), Natural Science Foundation of Shandong (No. BS2013SF009). }} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Shifeng Ding, Lihua Feng, Weijun Liu\\ \small Department of Mathematics\\[-0.8ex] \small Central South University\\[-0.8ex] \small Changsha 410083, China\\ \small\tt dingssff@163.com} % \date{\dateline{submission date}{acceptance date}\\ % \small Mathematics Subject Classifications: comma separated list of % MSC codes available from http://www.ams.org/mathscinet/freeTools.html} \date{\dateline{Dec 1, 2013}{Apr 4, 2014}{May 9, 2014}\\ \small Mathematics Subject Classifications: 05A19, 11M06} \begin{document} \maketitle % E-JC papers must include an abstract. The abstract should consist of a % succinct statement of background followed by a listing of the % principal new results that are to be found in the paper. The abstract % should be informative, clear, and as complete as possible. Phrases % like "we investigate..." or "we study..." should be kept to a minimum % in favor of "we prove that..." or "we show that...". Do not % include equation numbers, unexpanded citations (such as "[23]"), or % any other references to things in the paper that are not defined in % the abstract. The abstract will be distributed without the rest of the % paper so it must be entirely self-contained. \begin{abstract} Let $\zeta(s_1,s_2,\cdots,s_k;\alpha)$ be the multiple Hurwitz zeta function. Given two positive integers $k$ and $n$ with $k\leq n$, let $E(2n, k;\alpha)$ be the sum of all multiple zeta values with even arguments whose weight is $2n$ and whose depth is $k$. In this note we present some generating series for the numbers $E(2n,k;\alpha)$. % keywords are optional \bigskip\noindent \textbf{Keywords:}\ Multiple zeta values; Recursion algorithm; Generating series \end{abstract} \section{Introduction} The multiple zeta values \begin{equation}\zeta(s_1,s_2,\cdots,s_m)=\sum_{n_1>n_2>\cdots>n_k>0} \frac{1}{n_1^{s_1}n_2^{s_2}\cdots n_m^{s_m}},\nonumber \end{equation} are also called Euler-Zagier sums, where $s_1$, $s_2$, $\cdots$, $ s_m$ are positive integers with $s_1\geq2$. The multiple Hurwitz zeta function $\zeta(s_1,s_2,\cdots,s_k;\alpha)$ is defined by the multiple series \[\zeta(s_1,s_2,\cdots,s_k;\alpha)=\sum_{n_1>n_2>\cdots>n_k\geq0} \frac{1}{(n_1+\alpha)^{s_1}(n_2+\alpha)^{s_2}\cdots (n_k+\alpha)^{s_k}},\] where $\alpha\in\mathbb{C}\setminus\{0,-1,-2,\cdots\}$ and $s_1,s_2,\cdots,s_k$ are positive integers with $s_1>1$. The multiple zeta functions have attracted considerable interest in recent years. There are various relations among multiple zeta values. One of the well-known Q-linear relations among multiple zeta values is the sum formula\ ( see\ $\cite{Granville,Hoffman4,HoffmanAndMoen,Kaneko,Markett}$), which states that \[\sum_{\stackrel{j_1+\cdots+j_k=n}{\textnormal{\scriptsize {Each}}j_i\geq1,j_1>1}} \zeta(j_1,j_2,\cdots,j_k)=\zeta(n).\] In \cite{Masahiro Igarashi}, M. Igarashi proved a generalization of the sum formula. Here we quote Igarashi's result in a slightly different form. \begin{theorem}{\textnormal{(\cite{Masahiro Igarashi})}}\ \ Let $\alpha$ be a complex number with $\textnormal{Re}\ \alpha>0$ and let $n,k$ be positive integers such that $n\geq k+1$. Then the following identity holds. \begin{eqnarray}&&\sum_{\stackrel{j_1+j_2+\cdots+j_k=n}{\textnormal{Each}j_i\geq1, j_1\geq2}}\zeta(j_1,j_2,\cdots,j_k;\alpha)\nonumber\\ &=&\frac{1}{(n-k-1)!}\sum_{N=0}^\infty\frac{1}{(N+\alpha)^k}\left.\left[\frac{(1-x)_{N}} {(\alpha-x)_{N+1}}\right]^{(n-k-1)}\right|_{x=0},\nonumber \end{eqnarray} where $[\ \cdot \ ]^{(n-k-1)}$ denotes the $(n-k-1)$th derivative of a function with respect to its variable $x$, and $(a)_N$ denotes the Pochhammer symbol. \end{theorem} Given two positive integers $n$ and $k$ with $n\geq k$, let $E(2n, k;\alpha)$ be the sum of all the multiple zeta values of even-integer arguments having weight $2n$ and depth $k$, i.e., \[E(2n,k;\alpha)=\sum_{\stackrel{j_1+\cdots+j_k=n}{j_1,j_2,\cdots,j_k\geq1}} \zeta(2j_1,2j_2,\cdots,2j_k;\alpha).\] (The notation $E(2n,k;\alpha)$ is similar to that in \cite{Hoffman, Zhao}. In \cite{Hoffman}, Hoffman defined a number $E(2n,k)$ which is equivalent to the number $E(2n,k;1)$ in our notation, and in \cite{Zhao}, the number $E(2n,k;1/2)/2^{2n}$ in our notation is denoted by $T(2n,k)$.) The result of Gangl, Kaneko and Zagier \cite{Gangl} that \begin{eqnarray}E(2n,2;1)=\frac{3}{4}\zeta(2n), \ \ {\textnormal{for }}\ n\geq2.\nonumber \end{eqnarray} was extended by Shen and Cai \cite{Z.Y.Shen} to \begin{equation}E(2n,3;1)=\frac{5}{8}\zeta(2n)-\frac{1}{4}\zeta(2)\zeta(2n-2),\ \ \ {\textnormal{for }}n\geq3,\nonumber \end{equation} \begin{equation}E(2n,4;1)=\frac{35}{64}\zeta(2n)-\frac{5}{16}\zeta(2)\zeta(2n-2),\ \ \ {\textnormal{for }}n\geq4.\nonumber \end{equation} Applying the theory of symmetric functions, Hoffman \cite{Hoffman} and Zhao \cite{Zhao}, respectively, established the generating functions for the numbers $E(2n,k;1)$ and $T(2n,k)$, respectively. Hoffman \cite{Hoffman} proved that \[1+\sum_{n\geq k\geq1}E(2n,k;1)t^ns^k=\frac{\sin(\pi\sqrt{1-s}\sqrt{t})}{\sqrt{1-s}\sin(\pi\sqrt{t})}.\] And Zhao \cite{Zhao} proved that \[1+\sum_{n\geq k\geq1}T(2n,k)t^ns^k=\frac{\cos(\pi\sqrt{(1-s)t}/2)}{\cos(\pi\sqrt{t}/2)}.\] Based on these generating functions, some formulas for the numbers $E(2n,k;1)$ and $T(2n,k)$ for arbitrary $n\geq k$ have been obtained. For example, \begin{eqnarray}E(2n,k;1)=\frac{\zeta(2n)}{2^{2(k-1)}}{2k-1\choose k}- \sum_{j=1}^{[\frac{k-1}{2}]}\frac{\zeta(2j)\zeta(2n-2j)}{2^{2k-3}(2j+1)B_{2j}}{2k-2j-1\choose k}, \ \ \ \nonumber\end{eqnarray} \[T(2n,k)=\frac{t(2n)}{2^{2(k-1)}k}{2k-2\choose k-1}- \sum_{j=1}^{\lfloor\frac{k-1}{2}\rfloor}\frac{t(2j)t(2n-2j)}{2^{2k-3}(2^{2j}-1)kB_{2j}}{2k-2j-2\choose k-1},\] where $B_{2j}$ is the $2j$th Bernoulli number. In this note we use a method introduced by Granville \cite{Granville} to establish an algorithm similar to that in Theorem 1 for the calculation of the numbers $E(2n,k;\alpha)$. We prove that the calculation of $E(2n,k;\alpha)$ involves the Euler $\Gamma$-function and the direct formulas for $E(2n,k;1)$ and $E(2n,k;1/2)$ can be deduced from the Bessel functions of the first kind. The main results of this paper are the following theorems. \begin{theorem}\ Let $\alpha$ be a complex number with $\alpha\in\mathbb{C}\setminus\{0,-1,-2,\cdots\}$. Let $N\geq0$ denote an integer and let $(a_0^{(N)},a_1^{(N)},a_2^{(N)},\cdots)$ be an infinite series defined by the expansion \begin{equation}\prod_{\stackrel{r=0}{r\not=N}}^\infty\left(1+\frac{x}{(r+\alpha)^2-(N+\alpha)^2}\right)=a_0^{(N)}+a_1^{(N)}x+\cdots+a_{k-1}^{(N)}x^{k-1}+\cdots. \end{equation} Then for positive integers $n$ and $k$ (suppose $n\geq k$), we have \begin{equation}E(2n,k;\alpha)=\sum_{N=0}^\infty\frac{a_{k-1}^{(N)}}{(N+\alpha)^{2n-2k+2}}.\end{equation} \end{theorem} \begin{theorem}\ Given an integer $N\geq0$ and a complex number $\alpha\in\mathbb{C}\setminus\{0,-1,-2,\cdots\}$, for any $x\in\mathbb{C}$ we have \begin{eqnarray}&&\prod_{\stackrel{r=0}{r\not=N}}^\infty\left(1+\frac{x}{(r+\alpha)^2-(N+\alpha)^2}\right)\nonumber\\ &=&\frac{2(-1)^{N}(N+\alpha)\Gamma(2\alpha+N)}{N!}\frac{1}{x\Gamma(\alpha+\sqrt{(N+\alpha)^2-x})\Gamma(\alpha-\sqrt{(N+\alpha)^2-x})}, \nonumber \end{eqnarray} where $\Gamma$ denotes Euler's Gamma function. \end{theorem} \begin{corollary}\ Theorem 2 and Theorem 3 yield that: \begin{eqnarray}&&E(2n,k;\alpha)\nonumber\\ &=&\frac{2}{k!}\sum_{N=0}^\infty\frac{(-1)^N\Gamma(N+2\alpha)}{N!(N+\alpha)^{2n-2k+1}} \left.\left[\frac{1}{\Gamma(\alpha+\sqrt{(N+\alpha)^2-x}) \Gamma(\alpha-\sqrt{(N+\alpha)^2-x})}\right]^{(k)}\right|_{x=0},\nonumber \end{eqnarray} where $[\ \cdot \ ]^{(k)}$ denotes the $k$th derivative of a function with respect to its variable $x$ and\ $\Gamma$ denotes the Gamma function. \end{corollary} \begin{corollary}\ For positive integers $n $ and $k$ (suppose $n\geq k$ ), we have \begin{eqnarray}E(2n,k;1)&=&\frac{4(-1)^{k+1}\pi^{2k}}{k!}\sum_{N=0}^\infty \frac{(-1)^N}{(N+1)^{2n-2k}}\left(\cos\sqrt{x}\right)^{(k+1)}|_{x=(N\pi+\pi)^2},\\ E(2n,k;\frac{1}{2})&=&\frac{2(-1)^{k}\pi^{2k-1}}{k!}\sum_{N=0}^\infty \frac{(-1)^N}{(N+\frac{1}{2})^{2n-2k+1}}\left(\cos\sqrt{x}\right)^{(k)}|_{x=(N\pi+\pi/2)^2}. \end{eqnarray} \end{corollary} \begin{corollary}\ The direct formulas for $E(2n,k;1)$ and $E(2n,k;\frac{1}{2})$ are that: \begin{eqnarray}&&E(2n,k;1)=\frac{1}{2^{2k-2}k!} \sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-1)^{j}(2k-1-2j)!(2\pi)^{2j}}{(2j+1)!(k-1-2j)!}\cdot\zeta(2n-2j),\\ &&E(2n,k;\frac{1}{2})=\frac{1}{2^{2k-2}k!}\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-1)^j(2k-2-2j)!(2\pi)^{2j}}{(2j)!(k-1-2j)!} \zeta(2n-2j;\frac{1}{2}). \end{eqnarray} \end{corollary} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Some Lemmas} This paper uses some properties of Euler's Gamma function $\Gamma(s)$ and the Bessel function $J_{p}(x)$ of the first kind. Here we list some needed Lemmas. There are several equivalent forms of the Gamma function $\Gamma(s)$ developed by Euler, one by Weierstrass: \[\frac{1}{\Gamma(s)}=se^{\gamma s}\prod_{m=1}^\infty\left(1+\frac{s}{m}\right)e^{-\frac{s}{m}},\ \ \forall s\in\mathbb{C},\] where $\gamma$ denotes the Euler-Mascheroni constant defined by $$\gamma=\lim_{m\rightarrow\infty}(1+\frac{1}{2}+\cdots+\frac{1}{m}-\log m)=0.577215\cdots.$$ It is easy to deduce from Weierstrass' definition of $\Gamma(s)$ the following lemma. \begin{lemma}\ Let $\alpha_1,\alpha_2,\cdots,\alpha_k$ and $\beta_1,\beta_2,\cdots,\beta_k$ be two groups of complex numbers \textnormal{($k\geq1$)}. Suppose they satisfy the following conditions{\textnormal{:\ (i)}}\ $\alpha_1+\alpha_2+\cdots+\alpha_k=\beta_1+\beta_2+\cdots+\beta_k$ and{\textnormal{ (ii)}} none of the $\beta_j$s is a negative integer. Then we have \[\prod_{m=1}^\infty\frac{(m+\alpha_1)(m+\alpha_2)\cdots(m+\alpha_k)}{(m+\beta_1)(m+\beta_2)\cdots(m+\beta_k)} =\frac{\Gamma(1+\beta_1)\Gamma(1+\beta_2)\cdots\Gamma(1+\beta_k)}{\Gamma(1+\alpha_1)\Gamma(1+\alpha_2)\cdots\Gamma(1+\alpha_k)}. \] \end{lemma} It is well known that $\Gamma(s)$ satisfies the functional relation $\Gamma(s+1)=s\Gamma(s), \ \forall s\in\mathbb{C}\setminus\{0,-1,-2,\cdots\}$, and if $N\geq0$ is an integer then $\Gamma(s)$ has residue $\frac{(-1)^N}{N!}$ at $s=-N$. \begin{lemma}\textnormal{\cite{Remmert}} The following identities hold for $s\in\mathbb{C}$: \[\frac{1}{\Gamma(1+s)\Gamma(1-s)}=\frac{\sin\pi s}{\pi s},\ \ \ \ \ \ \ \frac{1}{\Gamma(\frac{1}{2}+s)\Gamma(\frac{1}{2}-s)}=\frac{\cos\pi s}{\pi } .\ \] \end{lemma} We turn to introduce some properties of the Bessel function of the first kind with a half-integer index. The Bessel function $J_p(x)$ is said to be of a half integer index if $p= k\pm\frac{1}{2}$ with $k$ being an integer. It is well known that the Bessel function of the first kind with a half integer index can be represented by elementary functions. \begin{lemma}\ \ $\frac{d^k}{dx^k}\cos\left(\sqrt{x}\right)=\sqrt{\frac{\pi}{2}}(-1)^k2^{-k}x^{\frac{1-2k}{4}} J_{k-\frac{1}{2}}(\sqrt{x}), \ \ \textnormal{for} \ x>0\ \textnormal{and}\ \ k=0,1,2,\cdots.$ \end{lemma} \begin{proof}\ The Bessel function $J_p(x)$ is defined by the series \begin{equation}J_p(x)=\left(\frac{x}{2}\right)^{p}\sum_{m=0}^\infty\frac{(-1)^m}{m!\Gamma(p+m+1)}\left(\frac{x}{2}\right)^{2m},\end{equation} where the radius of convergence of the series is $+\infty$. It follows from (7) that (using the well-known identity $\Gamma(n+1/2)=\sqrt{\pi}(2n)!/(n!2^{2n})$, $\forall n\geq0$) \begin{equation}J_{k-\frac{1}{2}}(\sqrt{x})=\sqrt{\frac{2}{\pi}}2^kx^{\frac{2k-1}{4}}\sum_{m=0}^\infty\frac{(-1)^m(m+k)!} {m!(2m+2k)!}x^m.\end{equation} On the other hand, by Taylor's expansion of $\cos\sqrt{x}$, we have \begin{align}\frac{d^k}{dx^k}\cos\left(\sqrt{x}\right) & =\sum_{m=k}^\infty\frac{(-1)^mm(m-1)\cdots(m-k+1)} {(2m)!}x^{m-k} \notag\\ &=\sum_{m=0}^\infty\frac{(-1)^{m+k}(m+k)!} {(2m+2k)!m!}x^{m}.\end{align} Observing the right-hand sides of (8) and (9), the result follows immediately. \end{proof} \begin{lemma}\ Let $k\geq0$ be an integer and let $x>0$. Then the Bessel function $J_{k+1/2}(x)$ is represented by \begin{eqnarray}J_{k+\frac{1}{2}}(x)&=&\sqrt{\frac{2}{\pi }}\frac{1}{\sqrt{x}}\left\{\sin x \sum_{j=0}^{\lfloor\frac{k}{2}\rfloor}\frac{(-1)^j(2k-2j)!}{(2j)!(k-2j)!}\cdot\frac{1}{(2x)^{k-2j}}\right.\nonumber\\ &&\left.\ \ \ \ \ \ \ \ \ \ \ \ -\cos x \sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-1)^j(2k-1-2j)!}{(2j+1)!(k-1-2j)!}\cdot\frac{1}{(2x)^{k-1-2j}}\right\}. \end{eqnarray} \end{lemma} \begin{proof}\ \ We omit the detailed steps. It is shown that\ (\cite{Y. Liu}) the Bessel function $J_{k+1/2}(x)$ satisfies the relation \begin{equation}J_{k+1/2}(x)=(-1)^k\sqrt{\frac{2}{\pi x}}x^{k+1}\frac{d^k}{(xdx)^k}\left(\frac{\sin x}{x}\right),\ \ \textnormal{for}\ \ k=0,1,2,\cdots.\end{equation} It follows from (11) that \[J_{1/2}=\sqrt{\frac{2}{\pi }}\frac{\sin x}{\sqrt{x}},\ \ \ \ \ J_{3/2}(x)=\sqrt{\frac{2}{\pi }}\frac{1}{\sqrt{x}}\left(\frac{\sin x}{x}-\cos x\right),\cdots,\] and so on. Then using induction on $k$ we can prove that (10) is equivalent to (11).\end{proof} \begin{lemma}Let $k\geq1$ be an integer. Then by Lemma 9 and Lemma 10, we have \begin{eqnarray}\left(\cos{\sqrt{x}}\right)^{(k)}&=&\frac{(-1)^{k}}{2^{2k-1}}\frac{\sin\sqrt{x}}{\sqrt{x}} \sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor}\frac{(-1)^j(2k-2-2j)!2^{2j}}{(2j)!(k-1-2j)!}\cdot\frac{1}{x^{k-1-j}}\nonumber\\ && -\frac{(-1)^{k}}{2^{2k-2}}\cos \sqrt{x} \sum_{j=0}^{\lfloor\frac{k-2}{2}\rfloor}\frac{(-1)^j(2k-3-2j)!2^{2j}}{(2j+1)!(k-2-2j)!}\cdot\frac{1}{x^{k-1-j}}. \end{eqnarray} \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ Proofs} %\textnormal{(}{\it Proof of Theorem 2 }\textnormal{)} \begin{proof}[Proof of Theorem 2]\ The left side of (2) is \begin{eqnarray} &&\sum_{\stackrel{j_1+\cdots+j_k=n}{j_1,\cdots,j_k\geq1}}\sum_{n_1>n_2>\cdots>n_k\geq0} \frac{1}{(n_1+\alpha)^{2j_1}(n_2+\alpha)^{2j_2}\cdots(n_k+\alpha)^{2j_k}}\nonumber\\ &&\ \ \ \ =\sum_{n_1>n_2>\cdots>n_k\geq0}\sum_{\stackrel{j_1+\cdots+j_k=n}{j_1,\cdots,j_k\geq1}} \frac{1}{(n_1+\alpha)^{2j_1}(n_2+\alpha)^{2j_2}\cdots(n_k+\alpha)^{2j_k}}.\end{eqnarray} The second sum in (13) is the coefficient of $x^{2n}$ in the formal power series \begin{eqnarray}&&\sum_{j=1}^\infty\left(\frac{x}{n_1+\alpha}\right)^{2j} \sum_{j=1}^\infty\left(\frac{x}{n_2+\alpha}\right)^{2j}\cdots\sum_{j=1}^\infty\left(\frac{x}{n_k+\alpha}\right)^{2j}\nonumber\\ &&\ \ =\frac{x^{2k}}{[(n_1+\alpha)^2-x^2][(n_2+\alpha)^2-x^2]\cdots[(n_k+\alpha)^2-x^2]}\nonumber\\ &&\ \ =\sum_{j=1}^{k}\left(\frac{x^{2k}}{(n_j+\alpha)^2-x^2}\prod_{\stackrel{1\leq m\leq k}{m\not=j}}\frac{1}{(n_m+\alpha)^2-(n_j+\alpha)^2}\right). \end{eqnarray} It follows that the coefficient of $x^{2n}$ above is \[\sum_{j=1}^k\left(\frac{1}{(n_j+\alpha)^{2n-2k+2}}\prod_{\stackrel{1\leq m\leq k}{m\not=j}}\frac{1}{(n_m+\alpha)^2-(n_j+\alpha)^2}\right).\] Hence the sum (13) is \begin{equation}\sum_{n_1>n_2>\cdots>n_k\geq0}\sum_{j=1}^k\left(\frac{1}{(n_j+\alpha)^{2n-2k+2}}\prod_{\stackrel{1\leq m\leq k}{m\not=j}}\frac{1}{(n_m+\alpha)^2-(n_j+\alpha)^2}\right). \end{equation} Now we take each $n_j=N$ in turn. Then the sum (15) becomes that \begin{equation}\sum_{N\geq0}\frac{1}{(N+\alpha)^{2n-2k+2}}\sum_{j=1}^k P(N,j-1)Q(N,k-j), \end{equation} where the series $\{P(N,i)\}_{i=0}^\infty$ and $\{Q(N,i)\}_{i=0}^\infty$ are defined in the following manner. Define $P(N,0)=Q(N,0)\triangleq1$, and $Q(N,j)=0$, $\forall j> N$; If $j>1$ then define $P(N,j-1)$ to be \begin{equation}P(N,j-1)=\sum_{n_1>\cdots>n_{j-1}>N}\prod_{i=1}^{j-1}\frac{1}{[(n_i+\alpha)^2-(N+\alpha)^2]};\nonumber\end{equation} While if $1\leq jn_{j+1}>\cdots>n_k\geq0}\prod_{i=j+1}^{k}\frac{1}{[(n_i+\alpha)^2-(N+\alpha)^2]}.\nonumber\end{equation} In other words, the series $\{P(N,i)\}_{i=0}^\infty$ and $\{Q(N,i)\}_{i=0}^\infty$ are defined by the following generating functions: \[1+P(N,1)x+P(N,2)x^2+\cdots=\prod_{r>N}\left(1+\frac{x}{(r+\alpha)^2-(N+\alpha)^2}\right),\] and \[1+Q(N,1)x+Q(N,2)x^2+\cdots=\prod_{0\leq r