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Multiple zeta values at non-positive integers. (English) Zbl 1002.11069
For Euler-Zagier’s multiple zeta function defined by \[ \zeta_k(s_1, s_2,\dots, s_k)=\sum_{0<n_1<n_2<\dots<n_k}{1\over n_1^{s_1} n_2^{s_2}\dots n_k^{s_k}} \] J. Zhao [Proc. Am. Math. Soc. 128, 1275-1283 (2000; Zbl 0949.11042)] gave an analytic continuation to a meromorphic function on all of \(\mathbb C^k\). The point \((-r_1, -r_2,\dots,-r_k)\) where \(r_i\) are nonnegative integers, lies on the set of singularities. Let the regular values of the multiple zeta function at nonpositive integers be defined as \[ \zeta_k(-r_1, -r_2,\dots,-r_k)=\lim_{s_1\to\;-r_1} \lim_{s_2\to\;-r_2}\dots\lim_{s_k\to\;-r_k}\zeta_k(s_1, s_2,\dots, s_k). \] In the paper under review the authors continue their research started with S. Egami in [Acta Arith. 98, 107-116 (2001; Zbl 0972.11085)]. They obtain that, for a nonnegative integer \(n\), \[ \begin{aligned} \zeta_k(0,\dots,0,-n) &=-{1\over(k-1)!(n+1)}\sum_{j=1}^k s(k,j) B_{n+j}+(-1)^k \delta_n,\\ \zeta_k(-n, 0,\dots,0) &={(-1)^n\over k!}\sum_{j=1}^ks(k,j){jB_{n+j}\over n+j}, \end{aligned} \] where \(\delta_n=1\) if \(n=0\) and \(\delta_n=0\) otherwise, \(s(k,j)\) are Stirling numbers of the first kind and \(B_n\) are Bernoulli numbers. Further they prove some symmetric formula where \(\zeta_k(-r_1, -r_2,\dots,-r_k)\) is expressed by a combination of \(\zeta_j\), \(j=1, 2,\dots, k-1\). The authors also consider the case when the values at nonpositive integers are defined in reverse order: \[ \lim_{s_k\to\;-r_k} \lim_{s_{k-1}\to\;-r_{k-1}}\dots\lim_{s_1\to\;-r_1}\zeta_k(s_1, s_2,\dots, s_k). \]

11M41 Other Dirichlet series and zeta functions
11B68 Bernoulli and Euler numbers and polynomials
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