Probabilistic degenerate Bernstein polynomials

TL;DR

This paper introduces probabilistic degenerate Bernstein polynomials that combine degeneracy and probabilistic extensions, deriving explicit formulas and identities, with special cases for common distributions.

Contribution

It presents a novel class of polynomials that unify degenerate and probabilistic Bernstein polynomials, expanding their theoretical framework.

Findings

Derived explicit expressions and identities for probabilistic degenerate Bernstein polynomials.

Explored special cases for Poisson, Bernoulli, and binomial random variables.

Provided new insights into the structure and properties of these polynomials.

Abstract

In recent years, both degenerate versions and probabilistic extensions of many special numbers and polynomials have been explored. For instance, degenerate Bernstein polynomials and probabilistic Bernstein polynomials were investigated earlier. Assume that Y is a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study probabilistic degenerate Bernstein polynomials associated with Y which are both probabilistic extension of the degenerate Bernstein polynomials and degenerate version of the probabilistic Bernstein polynomials associated with $Y$. We derive several explicit expressions and certain related identities for those polynomials. In addition, we treat the special cases of the Poisson random variable, the Bernoulli random variable and of the binomial random variable.