Exceptional Hahn and Jacobi orthogonal polynomials

TL;DR

This paper constructs new classes of exceptional Hahn and Jacobi orthogonal polynomials using Casorati and Wronskian determinants, extending classical families with eigenfunction properties and orthogonality under specific conditions.

Contribution

It introduces a novel method to generate exceptional Hahn and Jacobi polynomials via Casorati and Wronskian determinants, establishing their orthogonality and completeness.

Findings

Exceptional Hahn polynomials are orthogonal with respect to a positive measure.

Limit processes transform Hahn polynomials into exceptional Jacobi polynomials.

Conditions for orthogonality depend on parameters and admissibility criteria.

Abstract

Using Casorati determinants of Hahn polynomials $(h_n^{\alpha,\beta,N})_n$, we construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers polynomials $h_n^{\alpha,\beta,N;\F}$, $n\in \sigma _\F$, which are eigenfunctions of a second order difference operator, where $\sigma _\F$ is certain set of nonnegative integers, $\sigma _\F \varsubsetneq \NN$. When $N\in \NN$ and $\alpha$, $\beta$, $N$ and $\F$ satisfy a suitable admissibility condition, we prove that the polynomials $h_n^{\alpha,\beta,N;\F}$ are also orthogonal and complete with respect to a positive measure (exceptional Hahn polynomials). By passing to the limit, we transform the Casorati determinant of Hahn polynomials into a Wronskian type determinant of Jacobi polynomials $(P_n^{\alpha,\beta})_n$. Under suitable conditions for $\alpha$, $\beta$ and $\F$, these Wronskian type determinants turn out to be exceptional Jacobi polynomials.