On $q$-analogs of zeta functions associated with a pair of $q$-analogs of Bernoulli numbers and polynomials

TL;DR

This paper introduces two $q$-analogs of the Riemann zeta function using different methods and establishes their values at even integers in relation to $q$-Bernoulli and $q$-Euler numbers, extending classical zeta function theory.

Contribution

It presents novel $q$-analogs of the zeta function and links their special values to $q$-Bernoulli and $q$-Euler numbers, providing new insights into $q$-analogs of classical functions.

Findings

Values at even integers relate to $q$-Bernoulli and $q$-Euler numbers.

Two different approaches to define $q$-zeta functions.

Extension of classical zeta function properties to $q$-analogs.

Abstract

In this paper, we use two different approaches to introduce $q$-analogs of Riemann's zeta function and prove that their values at even integers are related to the $q$-Bernoulli and $q$ Euler's numbers introduced by Ismail and Mansour [Analysis and Applications, {\bf{17}}, 6, 2019, 853--895].