Approximation properties of q-Bernoulli polynomials

TL;DR

This paper investigates the approximation capabilities of q-Bernoulli polynomials, introduces a new q-operator related to the q-analogue of Euler-Maclaurin formula, and estimates functions using these polynomials with error analysis.

Contribution

It presents a novel q-operator and explores the approximation properties of q-Bernoulli polynomials, extending their application in differential equations and operator theory.

Findings

Derived a new q-operator for the q-analogue of Euler-Maclaurin formula

Provided error estimates for function approximation using q-Bernoulli polynomials

Highlighted applications in solving differential equations and operator theory

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 is 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 solving differential equations by estimating functions and second we may apply these techniques for operator theory.