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

TL;DR

This paper introduces a new generating function for (q-) Bernstein type polynomials, explores their properties, relations to other polynomials, and applications in approximation theory and statistics.

Contribution

It constructs a novel generating function for (q-) Bernstein polynomials and derives their properties, relations, and interpolation methods.

Findings

Derived recurrence relations and derivatives for (q-) Bernstein polynomials.

Established relations with Hermite, Bernoulli, and Stirling numbers.

Applied Mellin transform to define polynomial interpolation.

Abstract

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.