On an infinite number of nonlinear Euler sums

TL;DR

This paper extends the study of nonlinear Euler sums, specifically quadratic sums involving hyperharmonic numbers, providing explicit formulas in terms of zeta values and linear harmonic sums, thus broadening the understanding of these complex series.

Contribution

It generalizes previous results by including sums with mixed denominators and products of even and odd hyperharmonic numbers, revealing eight new families of expressible quadratic Euler sums.

Findings

Eight families of quadratic Euler sums expressed by zeta values and linear sums.

Sum formulas include mixed products of even and odd hyperharmonic numbers.

Results cover sums with various denominators, expanding known classes of Euler sums.

Abstract

Linear harmonic number sums had been studied by a variety of authors during the last centuries, but only few results are known about nonlinear Euler sums of quadratic or even higher degree. The first systematic study on nonlinear Euler sums consisting of products of hyperharmonic sums had been published by Flajolet and Salvy in 1997 followed by similar studies presented during the last years by different authors. Although these studies had been restricted to sums where the nominator consists of a product of even or odd hyperharmonic sums, where the denominator is of the type $1/k^n$. We have generalized these results to nonlinear Euler sums with different denominators and nominators which consist in addition of mixed products between even and odd hyperharmonic numbers. In detail we present eight families of quadratic Euler sums which are expressible by zeta values and special types of linear Euler sums only where the order of the nonlinear Euler sums is always an even number. The resulting eight different families of nonlinear Euler sums which we discovered consist of various products between even and odd hyperharmonic numbers, divided by three different types of denominators $1/k^n$, $1/((2k-1)^n)$ and $1/(k(2k-1))$. The calculational scheme is based on proper two-valued integer functions, which allow us to compute these sequences explicitly in terms of zeta values and pairs of odd-type linear harmonic numbers and even hyperharmonic numbers of second order.