Differential-difference properties of hypergeometric series

Abstract

Six families of generalized hypergeometric series in a variable x x and an arbitrary number of parameters are considered. Each of them is indexed by an integer n n . Linear recurrence relations in n n relate these functions and their product by the variable x 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 x . These formulas generalize well-known properties of the classical orthogonal polynomials.