On a Family of Hypergeometric Polynomials

TL;DR

This paper investigates a specific family of hypergeometric polynomials related to SCE problems, establishing their properties, relations, and representations, including Rodrigues formulas and generating functions, and connecting them to Laguerre polynomials with negative indices.

Contribution

The paper introduces and characterizes a new family of hypergeometric polynomials, deriving their Rodrigues formula, generating function, and expressing them via associated Laguerre polynomials with negative indices.

Findings

Polynomials are hypergeometric and distinguishable from classical ones.

Rodrigues formula for the polynomials is established.

Polynomials are expressed in terms of associated Laguerre polynomials with negative upper indices.

Abstract

We work on the SCE problems. We establish the expressions of three integrals' sequences, related to it, in terms of five families of polynomials. Relations between these integrals are demonstrated and we focus on one of the three problems : the determination of the family of polynomials noted $e_n (n \in \mathbb{N})$. We show taht these polynomials are hypergeometric. From this property, the NU method can be applied to this family. We have been able to determine the Rodrigues formula. These polynomials have properties that distinguish them from classical hypergeometric polynomials. We state and demonstrate the theorem adapted to the determination of the generating function of $e_n$. Finally, the sequence of polynomials studied is expressed in terms of associated Laguerre polynomials with negative upper indices.