Approximation properties of \(q\)-Bernoulli polynomials (Q1667583)

Summary: 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.