Functiones et Approximatio Commentarii Mathematici

Multinomial coefficients, multiple zeta values, Euler sums and series of powers of the Hurwitz zeta function

Michał Kijaczko

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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} $$ \displaystyle\sum_{n=1}^{\infty}\zeta_{H}^{m}(s,n)=\sum_{r=1}^{m}\sum_{|\boldsymbol{\alpha}|=m}{m\choose\boldsymbol{\alpha}}\zeta(\alpha_{1}s-1,\alpha_{2}s,\dots,\alpha_{r}s).$$

Article information

Source
Funct. Approx. Comment. Math., Volume 62, Number 2 (2020), 227-245.

Dates
First available in Project Euclid: 9 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.facm/1573268430

Digital Object Identifier
doi:10.7169/facm/1809

Mathematical Reviews number (MathSciNet)
MR4113987

Zentralblatt MATH identifier
07225511

Subjects
Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values 11M35: Hurwitz and Lerch zeta functions

Keywords
multiple zeta function Euler sum Riemann zeta function Hurwitz zeta function multinomial coefficients

Citation

Kijaczko, Michał. Multinomial coefficients, multiple zeta values, Euler sums and series of powers of the Hurwitz zeta function. Funct. Approx. Comment. Math. 62 (2020), no. 2, 227--245. doi:10.7169/facm/1809. https://projecteuclid.org/euclid.facm/1573268430


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References

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