On Generalized q-Poly-Bernoulli Numbers and Polynomials

Abstract

Many mathematicians in ([1], [2], [5], [14], [20]) introduced and investigated the generalized q-Bernoulli numbers and polynomials and the generalized q-Euler numbers and polynomials and the generalized q-Gennochi numbers and polynomials. Mahmudov ([15], [16]) considered and investigated the q-Bernoulli polynomials B-n,q((alpha)) in x; y of order alpha and the q-Euler polynomials B-n,q((alpha)) n;q (x, y) in x; y of order alpha. In this work, we define generalized q-poly-Bernoulli polynomials B-n,q([k,alpha]) (x, y) in x; y of order ff. We give new relations between the generalized q-poly-Bernoulli polynomials B-n,q([k,alpha]) (x, y)in n;q x; y in x; y of order ff and the generalized q-poly-Euler polynomials E[k, alpha] (x, y) in x; y of order ff and the q-Stirling numbers of the second kind S-2;q (n, k).