## Functiones et Approximatio Commentarii Mathematici

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

Michał Kijaczko

#### 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

https://projecteuclid.org/euclid.facm/1573268430

Digital Object Identifier
doi:10.7169/facm/1809

Mathematical Reviews number (MathSciNet)
MR4113987

Zentralblatt MATH identifier
07225511

#### 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

#### References

• [1] D. Borwein, J.M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. 38 (1995), 277–294.
• [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.
• [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.
• [5] A. Dil, I. Mezö and M. Cenkci, Evaluation of Euler-like sums via Hurwitz zeta values, Turk. J. Math. 41 (2017), 1640–1655.
• [6] P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics. 7(1) (1998), 15–35.
• [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.
• [9] C. Xu, Some evaluation of cubic Euler sums, arXiv:1705.06088.
• [10] C. Xu and J. Cheng, Some results on Euler sums, Functiones et Approximatio Commentarii Mathematici 54 (2016), 25–37.