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New Changhee \(q\)-Euler numbers and polynomials associated with \(p\)-adic \(q\)-integrals. (English) Zbl 1159.11049
Let \(p\) be an odd prime number. In this paper, the authors introduce \(q\)-analogues \(E_{n,q}(x)\) of the Euler polynomials and their multiple extensions \({E_{n,q}}^{(r)}(x\,|\,a_1,\ldots,a_r\,;\,b_1,\ldots,b_r)\) via \(p\)-adic integrals on \(\mathbb{Z}_p\). Here \(a_1,\ldots,a_r,b_1,\ldots,b_r\) are positive integers and \(q\) is suitably taken from \(\mathbb{C}\) or \(\mathbb{C}_p\); the \(p\)-adic completion of the algebraic closure of \(\mathbb{Q}_p\). Moreover, in the complex number field case, by calculating a generating function, the authors construct a multiple Hurwitz zeta function \(\zeta_{r}(s,x\,|\,a_1,\ldots,a_r\,;\,b_1,\ldots,b_r)\) which interpolates \({E_{n,q}}^{(r)}(x\,|\,a_1,\ldots,a_r\,;\,b_1,\ldots,b_r)\) at negative integer points. Namely, the equation \[ \zeta_{r}(-n,x\,|\,a_1,\ldots,a_r\,;\,b_1,\ldots,b_r) ={E_{n,q}}^{(r)}(x\,|\,a_1,\ldots,a_r\,;\,b_1,\ldots,b_r) \] holds for all odd positive integer \(n\). The authors also study a \(\chi\)-analogue of the above argument, where \(\chi\) is a Dirichlet character with odd conductor.

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11B68 Bernoulli and Euler numbers and polynomials
05A30 \(q\)-calculus and related topics
Full Text: DOI
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