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

Abstract

The main object of this paper is to construct a new generating function of the ($q\text{-}$) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the ($q\text{-}$) Bernstein-type polynomials. We also give relations between the ($q\text{-}$) 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 ($q\text{-}$) Bernstein-type polynomials. Moreover, we give some applications and questions on approximations of ($q\text{-}$) Bernstein-type polynomials, moments of some distributions in Statistics.