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

الملخص EN

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.