q-Hardy-Berndt type sums associated with q-Genocchi type zeta and l-functions

TL;DR

This paper introduces new q-analogues of Genocchi zeta and l-functions, constructs generating functions for q-Hardy-Berndt sums, and explores their relations, advancing the understanding of q-series and special functions.

Contribution

It defines novel q-analogues of classical functions and constructs new generating functions for q-Hardy-Berndt sums, linking them to Dirichlet characters and establishing new relations.

Findings

Defined q-analogue Genocchi zeta and l-functions

Constructed generating functions for q-Hardy-Berndt sums

Established new relations between q-Hardy-Berndt sums and q-Genocchi zeta function

Abstract

The aim of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, q-analogue Genocchi type l-function and two-variable q-Genocchi type l-function. Furthermore, we construct new genereting functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations related to q-Hardy-Berndt type sums and q-Genocchi zeta function as well.