General recurrence and ladder relations of hypergeometric-type functions

Abstract

A method for the explicit construction of general linear sum rules involving hypergeometric-type functions and their derivatives of any order is developed. This method only requires the knowledge of the coefficients of the differential equation that they satisfy, namely the hypergeometric-type differential equation. Special attention is paid to the differential-recurrence or ladder relations and to the fundamental three-term recurrence formulas. Most recurrence and ladder relations published in the literature for numerous special functions including the classical orthogonal polynomials, are instances of these sum rules. Moreover, an extension of the method to the generalized hypergeometric-type functions is also described, allowing us to obtain explicit ladder operators for the radial wave functions of multidimensional hydrogen-like atoms, where the varying parameter is the dimensionality.