Multivariate Interpolation Functions of Higher-Order q -Euler Numbers and Their Applications

Abstract

The aim of this paper, firstly, is to construct generating functions of $q$-Euler numbers and polynomials of higher order by applying the fermionic $p$-adic $q$-Volkenborn integral, secondly, to define multivariate $q$-Euler zeta function (Barnes-type Hurwitz $q$-Euler zeta function) and $l$-function which interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitz $q$-Euler zeta function and multivariate $q$-Euler $l$-function. Moreover, complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well.