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On the multiple \(q\)-Genocchi and Euler numbers. (English) Zbl 1192.11011

Summary: The purpose of this paper is to present a systematic study of some families of multiple \(q\)-Genocchi and Euler numbers by using the multivariate \(q\)-Volkenborn integral (= \(p\)-adic \(q\)-integral) on \(\mathbb Z_p\). The investigation of these \(q\)-Genocchi numbers and polynomials of higher order leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning ordinary \(q\)-Genocchi numbers and polynomials.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B65 Binomial coefficients; factorials; \(q\)-identities
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:

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