Abstract
In 2009, Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher-order, recently. In this paper, we extend our result to the higher-order twisted -Euler numbers and polynomials. The purpose of this paper is to establish various identities concerning higher-order twisted -Euler numbers and polynomials by the properties of -adic invariant integral on . Especially, if , we derive the result of Kim et al. (2009).
Highlights
Let p be a fixed odd prime number
Cp will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively
Let N be the set of natural numbers and Z normalized exponential valuation of Cp with |p|p p−vp p 1/p
Summary
Let p be a fixed odd prime number. Throughout this paper, the symbols Z, Zp, Qp, C, andCp will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively. Let us define the fermionic p-adic invariant integral on Zp as follows: pn −1 It is well known that the twisted q-Euler polynomials of order k are defined as ext 2 etζq k
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