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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
##### Keywords:
multiple zeta function; Euler sum
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##### References:
 [1] D. Borwein, J.M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. 38 (1995), 277-294. · Zbl 0819.40003 [2] K.-W. Chen, Multinomial sum formulas of multiple zeta values, arXiv:1704.05636. [3] J. Choi and H.M. Srivastava, Explicit evaluation of Euler and related sums, The Ramanujan Journal 10 (2005), 51-70. · Zbl 1115.11052 [4] J. Choi and H.M. Srivastava, The multiple Hurwitz zeta function and the multiple Hurwitz-Euler eta function, Taiwanese J. Math. 15 (2011), no. 2, 501-522. · Zbl 1273.11133 [5] A. Dil, I. Mezö and M. Cenkci, Evaluation of Euler-like sums via Hurwitz zeta values, Turk. J. Math. 41 (2017), 1640-1655. · Zbl 1424.11130 [6] P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics. 7(1) (1998), 15-35. · Zbl 0920.11061 [7] J.I.B. Gil and J. Fresán, Multiple zeta values: from numbers to motives, http://javier.fresan.perso.math.cnrs.fr/mzv.pdf. [8] K. Matsumoto, On the analytic continuation of various multiple zeta functions, in Number Theory for the Millennium II, M.A. Bennett et al. (eds.), A.K. Peters, Natick, 2002, pp. 417-440. · Zbl 1031.11051 [9] C. Xu, Some evaluation of cubic Euler sums, arXiv:1705.06088. · Zbl 1405.11111 [10] C. Xu and J. Cheng, Some results on Euler sums, Functiones et Approximatio Commentarii Mathematici 54 (2016), 25-37. · Zbl 1407.11097
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