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Evaluations of Euler-Type Sums of Weight \(\le 5\)

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Abstract

Let \(p,p_1,\ldots ,p_m\) be positive integers with \(p_1\le p_2\le \cdots \le p_m\) and \(x\in [-1,1)\), define the so-called Euler-type sums \({S_{{p_1}{p_2} \ldots {p_m},p}}\left( x \right) \), which are the infinite sums whose general term is a product of harmonic numbers of index n, a power of \(n^{-1}\) and variable \(x^n\), by
$$\begin{aligned} {S_{{p_1}{p_2} \ldots {p_m},p}}\left( x \right) : = \sum \limits _{n = 1}^{\infty } {\frac{{H_n^{\left( {{p_1}} \right) }H_n^{\left( {{p_2}} \right) } \ldots H_n^{\left( {{p_m}} \right) }}}{{{n^p}}}{x^n}}\quad (m\in \mathbb {N}:=\{1,2,3,\ldots \}), \end{aligned}$$
where \(H_n^{(p)}\) is defined by the generalized harmonic number. Extending earlier work about classical Euler sums, we prove that whenever \(p+p_1+\cdots +p_m \le 5\), then all sums \({S_{{p_1}{p_2} \ldots {p_m},p}}\left( 1/2\right) \) can be expressed as a rational linear combination of products of zeta values, polylogarithms and \(\log (2)\). The proof involves finding and solving linear equations which relate the different types of sums to each other.

Keywords

Harmonic number Polylogarithm function Euler sum Riemann zeta function Multiple zeta value Multiple harmonic sum 

Mathematics Subject Classification

11M06 11M32 11M99 

Notes

Acknowledgements

We are indebted to the two anonymous referees of the journal for their helpful remarks.

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenPeople’s Republic of China

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