A new generating function of (\(q\)-) Bernstein-type polynomials and their interpolation function. (English) Zbl 1185.33013

Summary: The main object of this paper is to construct a new generating function of the (\(q\)-) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relations and derivatives of the (\(q\)-) Bernstein-type polynomials. We also give relations between the (\(q\)-) Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and Stirling numbers of the second kind. By applying Mellin transformation to this generating function, we define interpolation of the (\(q\)-) Bernstein-type polynomials. Moreover, we give some applications and questions on approximations of (\(q\)-) Bernstein-type polynomials, and moments of some distributions in Statistics.


33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A15 Exact enumeration problems, generating functions
11B73 Bell and Stirling numbers
11B68 Bernoulli and Euler numbers and polynomials
41A10 Approximation by polynomials
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