On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

Dae Kim,Yonghong Yao,Taekyun Kim,Cheon Ryoo

doi:10.3390/sym10100451

Abstract

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

Highlights

We will consider p-adic integrals on Z p of any finite product of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials
After investigating some of their properties, we turned our attention to two variable q-Bernstein polynomials and operators, which was introduced by Kim and generalizes the single variable q-Bernstein polynomials and operators in [6]
We considered the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and showed that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials

Summary

Introduction

With the usual convention about replacing βnq by β n,q He defined q-Bernoulli polynomials as n n β n,q ( x ) = (q β q + [ x ]q ) = ∑. By considering p-adic integrals on Z p of them we can derive integral representations of q-Bernoulli numbers in the present paper, those of a q-analogue of Euler numbers in [5] and those of q-Euler numbers in [6]. We will study q-Bernoulli numbers and polynomials, which is introduced as p-adic invariant integrals on Z p , and investigate some properties for these numbers and polynomials. We will consider p-adic integrals on Z p of any finite product of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. Some identities for q-Bernoulli numbers will be found

Some Integral Representations of q-Bernoulli Numbers and Polynomials

Conclusions