Extensions of discrete classical orthogonal polynomials beyond the orthogonality

Abstract

It is well-known that the family of Hahn polynomials { h n α , β ( x ; N ) } n ≥ 0 is orthogonal with respect to a certain weight function up to degree N . In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can be characterized by a Δ -Sobolev orthogonality for every n and present a factorization for Hahn polynomials for a degree higher than N . We also present analogous results for dual Hahn, Krawtchouk, and Racah polynomials and give the limit relations among them for all n ∈ N 0 . Furthermore, in order to get these results for the Krawtchouk polynomials we will obtain a more general property of orthogonality for Meixner polynomials.