## Multivariate interpolation functions of higher-order $$q$$-Euler numbers and their applications.(English)Zbl 1140.11313

Summary: 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.

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)
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