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\(q\)-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, \({}_r\phi_s\)
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