On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials †

Lee-Chae Jang,Taekyun Kim,Dmitry Dolgy,Dae Kim

doi:10.3390/sym10080311

Lee-Chae Jang, Taekyun Kim + Show 2 more

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https://doi.org/10.3390/sym10080311

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Abstract

We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Zp and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Zp of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers.

Highlights

As is well known, the classical Bernstein polynomial of order n for f ∈ C [0, 1] is defined by, n kBn ( f | x ) = ∑ fBk,n ( x ), 0 ≤ x ≤ 1, (1)n k =0 where Bn is called the Bernstein operater of order n, and, Bk,n ( x ) =x (1 − x ) n − k, k n, k ≥ 0, (2)are called the Bernstein basis polynomials.The Weierstrass approximation theorem states that every continuous function defined on [0, 1]can be uniformly approximated as closely as desired by a polynomial function
We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Z p and investigate some properties for these numbers and polynomials
2. q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials

Summary

Introduction

The classical Bernstein polynomial of order n for f ∈ C [0, 1] is defined by (see [1,2,3]), N. Bernstein explicitly constructed a sequence of polynomials that uniformly approximates any given continuous function f on [0, 1]. For q ∈ C, with 0 < |q| < 1, and n, k ∈ Z≥0 , with n ≥ k, the q-Bernstein polynomials of degree n are defined by Kim as (see [8])

Results

Conclusion