Approximation Properties of q-Bernoulli Polynomials

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.