q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients

Abstract

The first purpose of this paper is to present a systemic study of some families of multiple q-Bernoulli numbers and polynomials by using multivariate q-Volkenborn integral (= p-adic q-integral) on Z p . From the studies of these q-Bernoulli numbers and polynomials of higher order we derive some interesting q-analogs of Stirling number identities.

Some identities of degenerate special polynomials

“…1/ N : (1) Then, by (1), we get I 1 .f 1 / D I .f / C 2f .0/ ; where f 1 .x/ D f .x C 1/ ; .see [9][10][11][12][13]…”

q-Bernoulli numbers and q-Bernoulli polynomials revisited

“…Note that lim q 1 [x] q = x. Since Carlitz brought out the concept of the q-extension of Bernoulli numbers and polynomials, many mathematicians have studied q-Bernoulli numbers and q-Bernoulli polynomials (see [1,7,5,6,[8][9][10][11][12]). Recently, Acikgöz, Erdal, and Araci have studied to a new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials (see [7]).…”

A new construction on the q-Bernoulli polynomials

“…Many mathematicians have studied the q-Bernoulli, q-Euler polynomials and related topics (see [1][2][3][4][5][6][7][8][9][10][11]). It is worth that Açikgöz et al [1] give a new approach to the q-Bernoulli polynomials and the q-Bernstein polynomials and show some properties.…”