A New Generating Function of (q‐) Bernstein‐Type Polynomials and Their Interpolation Function

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 relation and derivative of the ( q ‐) Bernstein‐type polynomials. We also give relations between the ( q ‐) Bernstein‐type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and the second‐kind Stirling numbers. 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, moments of some distributions in Statistics.