A Study on the p-Adic q-Integral Representation on Zp Associated with the Weighted q-Bernstein and q-Bernoulli Polynomials
نویسندگان
چکیده
Let p be a fixed prime number. Throughout this paper, p, p , and p will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of p , respectively. Let be the set of natural numbers, and let ∪ {0}. Let νp be the normalized exponential valuation of p with |p|p p−νp p 1/p. Let q be regarded as either a complex number q ∈ or a p-adic number q ∈ p . If q ∈ , then we always assume |q| < 1. If q ∈ p , we assume that |1 − q|p < 1. In this paper, we define the q-number as x q 1 − q / 1 − q see 1–13 . Let C 0, 1 be the set of continuous functions on 0, 1 . For α ∈ and n, k ∈ , the weighted q-Bernstein operator of order n for f ∈ C 0, 1 is defined by
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