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Twisted \((h,q)\)-Bernoulli numbers and polynomials related to twisted \((h,q)\)-zeta function and \(L\)-function. (English) Zbl 1139.11051

Summary: By using \(q\)-Volkenborn integral, we construct new generating functions of the new twisted \((h,q)\)-Bernoulli polynomials and numbers. By applying the Mellin transformation to these generating functions, we obtain integral representations of the new twisted \((h,q)\)-zeta function and twisted \((h,q)\)-\(L\)-function, which interpolate the twisted \((h,q)\)-Bernoulli numbers and generalized twisted \((h,q)\)-Bernoulli numbers at non-positive integers, respectively. Furthermore, relation between twisted \((h,q)\)-zeta function and twisted \((h,q)\)-\(L\)-function are proved. Some new relations, related to twisted \((h,q)\)-Bernoulli polynomials and numbers, are given.

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11B68 Bernoulli and Euler numbers and polynomials
11M41 Other Dirichlet series and zeta functions
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