Construction a new generating function of Bernstein type polynomials

TL;DR

This paper reconstructs a new generating function for Bernstein type polynomials, explores their properties, derivatives, recurrence relations, and interpolation functions, and discusses applications to Bézier curves.

Contribution

It introduces a novel generating function for Bernstein polynomials and derives new properties, relations, and applications not previously documented.

Findings

Derived the generating function for Bernstein type polynomials.

Established recurrence relations and derivative formulas.

Constructed an interpolation function via Mellin Transformation.

Abstract

Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also recurrence relations of these polynomials are found. Interpolation function of these polynomials is also constructed via Mellin Transformation. This function interpolates these polynomials at negative integers which are given explicitly. Moreover, relations between these polynomials, the generalized Stirling numbers, and Bernoulli polynomials of higher order are given. Furthermore some applications associated with B\'ezier curve are given.