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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:
 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, $${}_r\phi_s$$