Polynomial identities that involve binomial coefficients, double and rising factorials and their probabilistic interpretations and proofs

TL;DR

This paper presents polynomial identities involving binomial coefficients, factorials, and rising factorials, with probabilistic proofs, offering tools for simplifying complex expressions across various mathematical contexts.

Contribution

It introduces new polynomial identities with probabilistic proofs, expanding the toolkit for simplifying expressions involving factorials and binomial coefficients.

Findings

Identifies polynomial identities with simple and complex forms

Provides probabilistic proofs for these identities

Offers practical formulas for simplifying mathematical expressions

Abstract

We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The origins and the proofs of these identities are probabilistic. However, their form suggests universal applications in simplifying expressions. Many useful simplifying formulae are presented in the sequel.