Combinatorial sums, series and integrals involving odd harmonic numbers

TL;DR

This paper develops generating functions involving harmonic and odd harmonic numbers, evaluates related infinite series, and explores combinatorial identities and integrals with Fibonacci numbers, expanding the mathematical understanding of these special sums.

Contribution

It introduces new generating functions and closed-form evaluations for series involving harmonic, odd harmonic, and Catalan numbers, extending prior research.

Findings

Derived new generating functions involving harmonic numbers

Evaluated several infinite series in closed form

Established combinatorial identities with Catalan and Fibonacci numbers

Abstract

We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these generating functions we evaluate several infinite series in closed form. In addition, we offer some combinatorial sum identities involving Catalan numbers, harmonic numbers and odd harmonic numbers. Finally, we analyze a special log-integral with Fibonacci numbers and odd harmonic numbers.