On four families of power series involving harmonic numbers and central binomial coefficients

TL;DR

This paper introduces a novel summation technique to evaluate power series involving harmonic numbers and central binomial coefficients, leading to new formulas for nonlinear Euler sums expressed through zeta functions.

Contribution

It develops a special summation method using two-valued integer functions to compute series involving harmonic numbers and binomial coefficients, and relates these to nonlinear Euler sums and zeta functions.

Findings

Derived formulas for power series involving harmonic numbers and binomial coefficients.

Expressed nonlinear Euler sums with odd harmonic numbers in terms of zeta functions.

Provided a new computational approach for complex series involving harmonic numbers.

Abstract

We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate different families of power series which involve odd harmonic numbers and central binomial coefficients. Furthermore it is shown that based on these series a new type of nonlinear Euler sums that involve odd harmonic numbers can be calculated in terms of zeta functions.