Abstract
We investigate some interesting properties of the weighted -Bernstein polynomials related to the weighted -Bernoulli numbers and polynomials by using -adic -integral on .
Highlights
Introduction and PreliminariesÉ Let p be a fixed prime number
If q ∈ p, we assume that |1 − q|p < 1
The weighted q-Bernoulli numbers are constructed in previous paper 6 as follows: for α ∈ Æ, β0α,q
Summary
Introduction and PreliminariesÉ 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 Ép , respectively. The weighted q-Bernstein operator of order n for f ∈ C 0, 1 is defined by α n,q f |x n f k k0 n n k x k qα 1−x n−k q−α n f k0 k n Bkα,n x, q is called the weighted q-Bernstein polynomials of degree n see 2, 5, 6 . In 3 , Carlitz defined the expansion of Carlitz’s q-Bernoulli numbers as follows: β0h,q h h q
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