Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions

TL;DR

This paper introduces a novel method using functional equations and generating functions to derive and prove identities for Bernstein basis functions, including new formulas and series representations.

Contribution

It presents a new approach to deriving Bernstein polynomial identities through functional and differential equations, offering new proofs and identities.

Findings

New proofs for standard Bernstein identities

Derivation of new Bernstein identities

Series representations via Laplace transform

Abstract

The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions.