$q$-Bernoulli Numbers and Polynomials Associated with Multiple $q$-Zeta Functions and Basic $L$-series

TL;DR

This paper develops $p$-adic $q$-zeta functions and related $L$-series, introducing new generating functions for $q$-Bernoulli numbers and polynomials, and establishing their analytic properties and relations.

Contribution

It constructs novel $p$-adic $q$-zeta and $L$-functions, along with new generating functions, extending the theory of $q$-Bernoulli numbers and their interpolation properties.

Findings

Explicit formulas for $p$-adic $q$-zeta functions at negative integers

Analytic continuation of basic $q$-$L$-series

Relations between Barnes' type $q$-zeta and Changhee $q$-Bernoulli numbers

Abstract

By using $q$-Volkenborn integration and uniform differentiable on $\mathbb{Z}%_{p}$, we construct $p$-adic $q$-zeta functions. These functions interpolate the $q$-Bernoulli numbers and polynomials. The value of $p$-adic $q$-zeta functions at negative integers are given explicitly. We also define new generating functions of $q$-Bernoulli numbers and polynomials. By using these functions, we prove analytic continuation of some basic (or $q$-) $L$% -series. These generating functions also interpolate Barnes' type Changhee $% q $-Bernoulli numbers with attached to Dirichlet character as well. By applying Mellin transformation, we obtain relations between Barnes' type $q$% -zeta function and new Barnes' type Changhee $q$-Bernolli numbers. Furthermore, we construct the Dirichlet type Changhee (or $q$-) $L$% -functions.