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A new extension of \(q\)-Euler numbers and polynomials related to their interpolation functions. (English) Zbl 1152.11009

Summary: In this work, by using a \(p\)-adic \(q\)-Volkenborn integral, we construct a new approach to generating functions of the \((h,q)\)-Euler numbers and polynomials attached to a Dirichlet character \(\chi \). By applying the Mellin transformation and a derivative operator to these functions, we define \((h,q)\)-extensions of zeta functions and \(l\)-functions, which interpolate \((h,q)\)-extensions of Euler numbers at negative integers.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11S40 Zeta functions and \(L\)-functions
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References:

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