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

Abstract

By using $q$-Volkenborn integration and uniform differentiable on $\mathbb{Z}$, 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 -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.

37 pages

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• QKZ equation with |q|=1 and correlation functions of the XXZ model in the gapless regime
• Sums of powers of consecutive q-integers
• $p$-adic discrete dynamical systems and their applications in physics and cognitive sciences

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