Evaluations of Euler type sums of weight $\le$ 5

Abstract

Let $p,p_1,\dots ,p_m$ 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 $S_{p_1{p}_2\cdots p_m,p}\left(x\right)$, 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 \[S_{p_1 p_2 \cdots p_m, p}(x) := \sum_{n = 1}^\infty \frac{H_n^{(p_1)} H_n^{(p_2)} \cdots H_n^{(p_m)}} {n^p} x^n \quad (m\in \mathbb{N} := \{1,2,3,\ldots\}), \] 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 $S_{p_1{p}_2\cdots p_m,p}\left(1/2\right)$ 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.

References in corpus (5)

• Multiple zeta values and Euler sums
• On harmonic numbers and nonlinear Euler sums
• Explicit evaluation of harmonic sums
• Integrals of logarithmic functions and alternating multiple zeta values
• Identities for the multiple zeta (star) values

Cited by in corpus (3)

• On logarithmic integrals, harmonic sums and variations
• Two applications of polylog functions and Euler sums
• Evaluations of multiple polylogarithm functions, multiple zeta values and related zeta values