Degenerate Bernstein Polynomials

TL;DR

This paper introduces degenerate Bernstein polynomials, exploring their properties and connections to other degenerate polynomials, motivated by recent work on multidimensional Bernstein polynomials and degenerate Bernoulli polynomials.

Contribution

It derives generating functions, identities, and recurrence relations for degenerate Bernstein polynomials, establishing links with generalized factorial and Bernoulli polynomials.

Findings

Derived generating function and symmetric identities

Established recurrence relations

Connected degenerate Bernstein polynomials with other degenerate polynomials

Abstract

Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and their applications'([15,16]) and Carlitz's degenerate Bernoulli polynomials. We derived thier generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.