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

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].