\(q\)-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. (English) Zbl 1196.11040
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, \({}_r\phi_s\) |
