\(q\)-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. (English) Zbl 1196.11040

Kim, Taekyun

Russ. J. Math. Phys. 15, No. 1, 51-57 (2008).

Summary: A purpose of this paper is to present a systemic study of some families of multiple \(q\)-Bernoulli numbers and polynomials by using the multivariate \(q\)-Volkenborn integral (= \(p\)-adic \(q\)-integral) on \(\mathbb Z_p\) . Moreover, the study of these higher-order \(q\)-Bernoulli numbers and polynomials implies some interesting \(q\)-analogs of Stirling number identities.

MSC:

11B68	Bernoulli and Euler numbers and polynomials
11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
05A10	Factorials, binomial coefficients, combinatorial functions
05A30	\(q\)-calculus and related topics
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Keywords:

multiple \(q\)-Bernoulli numbers; multiple \(q\)-Bernoulli polynomials; multivariate \(q\)-Volkenborn integral; \(q\)-analogs of Stirling number identities