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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.

11S80Other analytic theory of local fields
11B68Bernoulli and Euler numbers and polynomials
33D15Basic hypergeometric functions of one variable, ${}_r\phi_s$