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\(q\)-Volkenborn integration. (English) Zbl 1092.11045
Summary: The “complete sum for \(q\)-Bernoulli polynomials” is evaluated by using the \(q\)-Volkenborn integral. This sum helps us to treat the relationships between \(q\)-Volkenborn integral and non-Archimedean combinatorial analysis. Also \(q\)-analogs of Stirling number identities are formulated, and the interconsistency among the \(q\)-analogs of the Stirling numbers and of the Bernoulli numbers is investigated; \(p\)-adic valued \(q\)-Bernoulli distributions and related probability properties of Bernoulli measures are studied in connection with a stochastic biological model.

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11B68 Bernoulli and Euler numbers and polynomials
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)