Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme

TL;DR

This paper introduces a new type of forward and backward shift relations for hypergeometric orthogonal polynomials in the Askey scheme, focusing on parameter shifts rather than variable shifts.

Contribution

It presents novel shift relations derived from alternative factorizations, expanding the understanding of parameter transformations in orthogonal polynomials.

Findings

New shift relations that only shift parameters

Relations applicable to various polynomials in the Askey scheme

Enhanced factorization approach for deriving shift relations

Abstract

The forward and backward shift relations are basic properties of the (basic) hypergeometric orthogonal polynomials in the Askey scheme (Jacobi, Askey-Wilson, $q$-Racah, big $q$-Jacobi etc.) and they are related to the factorization of the differential or difference operators. Based on other factorizations, we obtain another type of forward and backward shift relations. Essentially, these shift relations shift only the parameters.