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An analogue of the zeta function and its applications. (English) Zbl 1112.11013
Summary: We construct new analogues of Bernoulli numbers and polynomials. We define the \(q\)-extension of zeta function. Finally we give relation between the \(q\)-extension of zeta functions and the \((h,q)\)-extension of Bernoulli polynomials.

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11S40 Zeta functions and \(L\)-functions
33B30 Higher logarithm functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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[1] Kim, T., Analytic continuation of multiple \(q\)-zeta functions and their values at negative integers, Russ. J. math. phys, 11, 71-76, (2004) · Zbl 1115.11068
[2] Kim, T., Non-Archimedean \(q\)-integrals associated with multiple changhee \(q\)-Bernoulli polynomials, Russ. J. math. phys., 10, 91-98, (2003) · Zbl 1072.11090
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[4] Kim, T., \(q\)-Riemann zeta functions, Int. J. math. math. sci., 2004, 12, 599-605, (2004) · Zbl 1122.11082
[5] Kim, T.; Rim, S.H., Generalized carlitz’s \(q\)-Bernoulli numbers in the \(p\)-adic number field, Adv. stud. contemp. math., 2, 9-19, (2000) · Zbl 1050.11020
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