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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.

##### MSC:
 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
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