Some possible $q$-generalizations of harmonic numbers

TL;DR

This paper explores three different $q$-analogues of harmonic numbers, providing generating functions, and linking them to $q$-gamma and $q$-digamma functions, with applications in number theory.

Contribution

Introduces new $q$-generalizations of harmonic numbers and derives related generating functions and connections to special functions.

Findings

Derived generating functions involving number theoretical functions

Presented a $q$-generalization of Gosper's exponential generating function

Connected $q$-harmonic numbers to $q$-gamma and $q$-digamma functions

Abstract

We study three different $q$-analogues of the harmonic numbers. As applications, we present some generating functions involving number theoretical functions and give the $q$-generalization of Gosper's exponential generating function of harmonic numbers. We involve also the $q$-gamma and $q$-digamma function.