# Evaluations of Euler-Type Sums of Weight $$\le 5$$

Article

## 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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