Finite differences and orthogonal polynomials

Abstract

Abstract: Explicit representations of specific sets of orthogonal polynomials are often not as useful as one would like them to be. However, being able to work with them can be useful. There is a set of polynomials found by Karlin and McGregor and by Carlitz at the same time. The representation they found as a hypergeometric series shows these are at least rational functions. I can now show directly from the hypergeometric representation why they are polynomials. A similar argument is used to obtain the continuous q-Hermite polynomials as a limit from more general orthogonal polynomials.

Keywords

• finite differences
• orthogonal polynomials