Abstract
Recently, Ryoo introduced the weighted q-Euler numbers and polynomials which are a slightly different Kim's weighted q-Euler numbers and polynomials(see C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, 2011]). In this paper, we give some interesting new identities on the weighted q-Euler numbers related to the q-Bernstein polynomials
Highlights
Let p be a fixed odd prime number
When one talks of q-extension, q is variously considered as an indeterminate, a complex number q Î C, or a p-adic number q Î Cp
In this paper we study the weighted q-Bernstein polynomials to express the fermionic q-integral on Zp and investigate some new identities on the weighted q-Euler numbers related to the weighted q-Bernstein polynomials
Summary
Let p be a fixed odd prime number. Throughout this paper Zp, Qp, C and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number fields and the completion of algebraic closure of Qp, respectively. Let f be a continuous function on Zp. For a Î N and k, n Î Z+, the weighted p-adic q-Bernstein operator of order n for f is defined by Kim as follows: n [x]kqα [1 − x]nq−−αk are called the q-Bernstein polynomials of degree n with weighted a. X = 0, E(nα,q)(0) = E(nα,q) are called the n-th q-Euler numbers with weight a (see [14]).
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