## $$q$$-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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