On the q-Genocchi Numbers and Polynomials with Weight α and Weak Weight β

نویسندگان

Jung Yoog Kang

H. Y. Lee

Nam Soon Jung

چکیده

The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the q-Genocchi numbers and polynomials see 1–13 . In this paper, we construct a new type of q-Genocchi numbers G α,β n,q and polynomials G α,β n,q x with weight α and weak weight β. Throughout this paper, we use the following notations. By Zp, we denote the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, Cp denotes the completion of algebraic closure of Qp, N denotes the set of natural numbers, Z denotes the ring of rational integers, Q denotes the field of rational numbers, C denotes the set of complex numbers, and Z N ∪ {0}. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1. When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C, one normally assume that |q| < 1. If q ∈ Cp, we normally assumes that |q − 1|p < p− 1/p−1 so that q exp x log q for |x|p ≤ 1. Throughout this paper, we use the notation

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