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Multinomial coefficients, multiple zeta values, Euler sums and series of powers of the Hurwitz zeta function. (English) Zbl 07225511
The multiple zeta function is defined by \[ \zeta(s_1,\dots, s_k)=\sum_{1\leq n_1<\cdots <n_k}\frac{1}{n_1^{s_1}\cdots n_k^{s_k}} \] for \(\mathfrak{Re}(s_{j-i+1})+\cdots+\mathfrak{Re}(s_j)>i, 1\leq i\leq j\). Chen’s theorem says that \[ \sum_{r=1}^m\sum_{|\overrightarrow{\alpha}|=m}\binom{m}{\overrightarrow{ \alpha}}\zeta(n \overrightarrow{\alpha})=\zeta^m(n) \] for \(m\geq 1, n\geq 2\), where \(\overrightarrow{\alpha}=(\alpha_1,\dots, \alpha_r), \binom{m}{\overrightarrow{\alpha}}=\frac{m!}{\alpha_1!\cdots \alpha_r!}\).
The multiple Hurwitz zeta function is defined by \[ \zeta_H(s_1,\dots, s_k;a)=\sum_{1\leq n_1<\cdots<n_k}\frac{1}{(n_1+a)^{s_1}\cdots (n_k+a)^{s_k}} \] for \(\mathfrak{Re}(s_{j-i+1})+\cdots+\mathfrak{Re}(s_j)>i, 1\leq i\leq j\) and \(a>0\). Similarly, the multiple Lerch zeta function is defined by \[ \Phi(z_1,\dots, z_k; s_1,\dots, s_k;a)=\sum_{1\leq n_1<\cdots<n_k}\frac{z_1^{n_1}\cdots z_k^{n_k}}{(n_1+a)^{s_1}\cdots (n_k+a)^{s_k}} \] for \(\mathfrak{Re}(s_{j-i+1})+\cdots+\mathfrak{Re}(s_j)>i, 1\leq i\leq j\) , \(|z_l|\leq 1, l=1,\dots,k\) and \(a>0\).
In this paper, the author generalizes Chen’s theorem to the multiple Hurwitz zeta function and multiple Lerch zeta function. Furthermore, the author also gives some formulas about Euler sums and generating functions of powers of Hurwitz zeta function.
MSC:
11M32 Multiple Dirichlet series and zeta functions and multizeta values
11M35 Hurwitz and Lerch zeta functions
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References:
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