General recurrence and ladder relations of hypergeometric-type functions

  • Authors:

  • A. Zarzo;R. J. Yáñez;J. S. Dehesa

  • Affiliations:

  • Departamento de Matemática Aplicada, E.T.S. Ingenieros Industriales, Universidad Politécnica de Madrid, Spain and Instituto Carlos I de Física Teórica y Computacional, Facultad ...;Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain and Instituto Carlos I de Física Teórica y Computacional, Facultad de Cienc ...;Departamento de Física Moderna, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain and Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias, ...

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2007

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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.