Some Identities on the q - Bernoulli Numbers and Polynomials with Weight 0

and Applied Analysis 3 where n, k ∈ Z see 1, 9, 10 . For n, k ∈ Z , the p-adic Bernstein polynomials of degree n are defined by Bk,n x k x k 1 − x n−k for x ∈ Zp, see 1, 10, 11 . In this paper, we consider Bernstein polynomials to express the p-adic q-integral on Zp and investigate some interesting identities of Bernstein polynomials associated with the q-Bernoulli numbers and polynomials with weight 0 by using the expression of p-adic qintegral on Zp of these polynomials. 2. q-Bernoulli Numbers with Weight 0 and Bernstein Polynomials In the special case, α 0, the q-Bernoulli numbers with weight 0 will be denoted by β̃ 0 n,q β̃n,q. From 1.4 , 1.5 , and 1.6 , we note that ∞ ∑ n 0 β̃n,q t n! ∞ ∑ n 0 ∫