General Linear Transformations of Hypergeometric Functions

Abstract

The notion of a canonical form is introduced for multiple hypergeometric series. This notion, in conjunction with the factorization method suggested earlier by the author, is used to obtain the most general explicit descriptions of linear transformations of multiple series that are of Gauss, Kummer, or Bessel type with respect to some variable \(x_n \). A complete classification of the 34 Horn series according to their types and forms is given. It is used to show that the transformations described in this paper permit one to obtain the 147 single transformations of Horn series as well as quite a few repeated transformations. A computer program implementing these transformations is developed on the basis of the Maple V-4 computer algebra system.