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June 2020 Multinomial coefficients, multiple zeta values, Euler sums and series of powers of the Hurwitz zeta function
Michał Kijaczko
Funct. Approx. Comment. Math. 62(2): 227-245 (June 2020). DOI: 10.7169/facm/1809

Abstract

We generalize Chen's theorem [2] $$\displaystyle\sum_{r=1}^{m}\sum_{|\boldsymbol{\alpha}|=m}{m\choose\boldsymbol{\alpha}}\zeta(\boldsymbol{\alpha}n)=\zeta^{m}(n)$$ for complex arguments, presenting a very elementary proof. Subsequently, using a similar technique, we obtain a general formula that allows us to prove relations between nonlinear Euler sums. We also present relations between series of powers of the Hurwitz zeta function and the multiple zeta function, like\vspace{-1pt} $$ \displaystyle\sum_{n=1}^{\infty}\zeta_{H}^{m}(s,n)=\sum_{r=1}^{m}\sum_{|\boldsymbol{\alpha}|=m}{m\choose\boldsymbol{\alpha}}\zeta(\alpha_{1}s-1,\alpha_{2}s,\dots,\alpha_{r}s).$$

Citation

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Michał Kijaczko. "Multinomial coefficients, multiple zeta values, Euler sums and series of powers of the Hurwitz zeta function." Funct. Approx. Comment. Math. 62 (2) 227 - 245, June 2020. https://doi.org/10.7169/facm/1809

Information

Published: June 2020
First available in Project Euclid: 9 November 2019

zbMATH: 07225511
MathSciNet: MR4113987
Digital Object Identifier: 10.7169/facm/1809

Subjects:
Primary: 11M32, 11M35

Rights: Copyright © 2020 Adam Mickiewicz University

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Vol.62 • No. 2 • June 2020
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