Differential-Difference Properties of Hypergeometric Series

TL;DR

This paper investigates the differential-difference properties of hypergeometric series, deriving explicit recurrence relations and factorizations that extend classical orthogonal polynomial properties.

Contribution

It introduces new factorizations of recurrence relations for hypergeometric series with multiple parameters, generalizing known properties of classical orthogonal polynomials.

Findings

Explicit factorizations of recurrence relations as products of first order operators

Recurrences derived for derivatives with respect to x

Generalization of classical orthogonal polynomial properties

Abstract

Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product by the variable $x$. We give explicit factorizations of these equations as products of first order recurrence operators. Related recurrences are also derived for the derivative with respect to $x$. These formulas generalize well-known properties of the classical orthogonal polynomials.