Extensions of discrete classical orthogonal polynomials beyond the orthogonality

TL;DR

This paper extends the understanding of classical discrete orthogonal polynomials by exploring their properties beyond traditional orthogonality, including factorization, Sobolev orthogonality, and limit relations among various polynomial families.

Contribution

It introduces a factorization for Hahn polynomials beyond degree N and characterizes these and other polynomials via $ riangle$-Sobolev orthogonality, expanding their theoretical framework.

Findings

Factorization of Hahn polynomials for degrees higher than N

Characterization of polynomials via $ riangle$-Sobolev orthogonality

Limit relations between Hahn, dual-Hahn, Krawtchouk, and Racah polynomials

Abstract

It is well known that the family of Hahn polynomials $\{h_n^{\alpha,\beta}(x;N)\}_{n\ge 0}$ is orthogonal with respect to a certain weight function up to $N$. In this paper we present a factorization for Hahn polynomials for a degree higher than $N$ and we prove that these polynomials can be characterized by a $\Delta$-Sobolev orthogonality. We also present an analogous result for dual-Hahn, Krawtchouk, and Racah polynomials and give the limit relations between them for all $n\in \XX N_0$. Furthermore, in order to get this results for the Krawtchouk polynomials we will get a more general property of orthogonality for Meixner polynomials.