Approximation Properties of q-Bernoulli Polynomials

M Momenzadeh,I Y Kakangi

doi:10.1155/2017/9828065

Abstract

We study the q-analogue of Euler-Maclaurin formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of q-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.

Highlights

The present study has sought to investigate the approximation of suitable function f(x) as a linear combination of q-Bernoulli polynomials
This study applies q-operator to expand a function in terms of q-Bernoulli polynomials. This method leads us to q-analogue of Euler expansion
We approximate f(x) as a linear combination of q-Bernoulli polynomials and we assume that f(x) ≃ ∑Nn=0 Cnβn,q(x), so taking the Jackson integral from both sides leads us to

Summary

Introduction

The present study has sought to investigate the approximation of suitable function f(x) as a linear combination of q-Bernoulli polynomials. This study applies q-operator to expand a function in terms of q-Bernoulli polynomials. A kind of approximation of a function in terms of Bernoulli polynomials is used in several approaches for solving differential equations, such as [2,3,4]. This paper gives conditions to approximate capable functions as a linear combination of q-Bernoulli polynomials as well as related examples.

Results

Conclusion