Some identities on the q-Genocchi polynomials with weak weight \alpha and Bernstein polynomials

Abstract

Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z+ = N ∪ {0}. The padic absolute value is defined by |x|p = 1 pr , where x = p s t ( r ∈ Q and s, t ∈ Z with (s, t) = (p, s) = (p, t) = 1). In this paper we assume that q ∈ Cp with |q − 1|p < 1 as an indeterminate. The q-number is defined by