Power series with skew-harmonic numbers, dilogarithms, and double integrals

TL;DR

This paper evaluates power series involving skew-harmonic numbers, dilogarithms, and double integrals, providing closed-form solutions and addressing recent open problems in mathematical series and constants.

Contribution

It introduces new closed-form evaluations of series with skew-harmonic numbers and connects them to dilogarithms, solving recent open problems and expanding the understanding of related integrals.

Findings

Closed-form solutions for series with skew-harmonic numbers

Connections between series and dilogarithm/trilogarithm functions

Evaluations of double integrals in terms of classical constants

Abstract

The skew-harmonic numbers are the partial sums of the alternating harmonic series, i.e. the expansion of log(2). We evaluate in closed form various power series and numerical series with skew-harmonic numbers. This provides a simultaneous solution of two recent problems by Ovidiu Furdui in the American Mathematical Monthly and the College Mathematics Journal. We also present and discuss representations involving the dilogarithm and the trilogaithm which are related to our results. Finally, we provide the evaluations of several double integrals in terms of classical constants.