On the Kummer Solutions of the Hypergeometric Equation

Abstract

One of the oldest, and still one of the most interesting, applications of group theory arises in the study of the transformations of an ordinary differential equation. If we know that a given differential equation admits a group of transformations, then we know that the solution set must admit that same group of transformations, and we can deduce properties of all the solutions from the properties of any one of them. A case in point is offered by the celebrated hypergeometric equation (See Eq. (1) below), whose solutions include many of the most interesting special functions of mathematical physics. In his book [3], Einar Hille notes that this equation has a venerable history associated with such names as Gauss, Euler, Riemann, and Kummer. The hypergeometric equation is in fact a prototype: every ordinary differential equation of second order with at most three regular singular points can be brought to the hypergeometric equation by means of suitably chosen changes of variable [S]. In 1836 Kummer published a set of 6 distinct solutions of the hypergeometric equation. These include the hypergeometric function of Gauss, and all of them could be expressed in terms of Gauss's function (See Table 1 below). A useful summary of their basic properties is found in [1, p. 105ff.]. A glance at the list of these Kummer solutions reveals a rather complicated set of relationships which pleads for some simple explanation. We show here that the Kummer solutions are related by a finite group of transformations which serve to explain their relationships and to exemplify the use of transformation groups in the study of differential equations.