A new generating function of (\(q\)-) Bernstein-type polynomials and their interpolation function (Q963189)

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.