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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.

MSC:
11S80Other analytic theory of local fields
11B68Bernoulli and Euler numbers and polynomials
33D15Basic hypergeometric functions of one variable, r ϕ s