Abstract
In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.
Highlights
1 Introduction Let p be chosen as a fixed odd prime number
When one talks of a q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp
The following measure is defined by Kim: for any positive integer n and ≤ a < pn, μq a + pnZp
Summary
Let p be chosen as a fixed odd prime number. In this paper Zp, Qp, C and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex numbers, and the completion of an algebraic closure of Qp.Let vp be a normalized exponential valuation of Cp by |p|p = p–vp (p) pWhen one talks of a q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp. The following measure is defined by Kim: for any positive integer n and ≤ a < pn, μq a + pnZp Extended q-Euler polynomials ( known as weighted q-Euler polynomials) are defined by In the case x = in ( ), we have En(α,q)( ) := En(α,q), which are called extended q-Euler numbers (or weighted q-Euler numbers).
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