Lie Theory and Generalizations of the Hypergeometric Functions

Abstract

In this paper we use the differential recurrence relations satisfied by the ${}_2 F_1 $ and their generalizations the${}_p F_q $ and Lauricella functions to associate a Lie algebra (dynamical symmetry algebra) with each of these families of special functions. We demonstrate that the representation theory of the Lie algebras yields a variety of addition theorems and generating functions for the associated families. Most of the results of this survey are contained in the author’s papers [1]–[3] but the comments on the functions $F_A $, $F_B $, $F_C $ are new.