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Approximation properties of \(q\)-Bernoulli polynomials. (English) Zbl 1470.41005

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.

MSC:

41A10 Approximation by polynomials
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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