On the Solution of Linear Homogeneous Differential Equations

Abstract

A standard way of solving linear homogeneous differential equations with polynomial coefficients is the series method of Frobenius. The differential equations of the generalized hypergeometric functions, when solved by this method, have non logarithmic series solutions whose coefficients are found directly by solving a two-term recurrence relation as well as logarithmic solutions that are derivable from a two-term recurrence relation. We are here interested in a converse of this statement: THEOREM: If the non-logarithmic solutions of a linear homogeneous differential equation with polynomial coefficients are series whose coefficients are solutions of a two-term recurrence relation, then the differential equation is essentially a generalized hypergeometric differential equation and the solutions are generalized hypergeometric series or logarithmic solutions associated with generalized hypergeometric serses. This theorem is well-known to workers in the special functions of analysis but since it has seemed to be such a surprising result to others, we believe that this theorem should be made a matter of public record. Recurrence relations with more than two terms are so difficult to solve explicitly that virtually all the examples of the Frobenius method given in the standard textbooks that yield series with explicitly given coefficients are examples covered by this theorem. Our analysis will rest heavily on the use of the operator O=x(d/dx). It is a linear operator having the properties Oxn = nXn and 0 [on ]f(X) =-on+lf(X). Let { Pk(6) }I = be a set of polynomials in 0 with coefficients independent of x and such that the polynomials of greatest degree have degree precisely t. Then the most general linear homogeneous differential equation with polynomial coefficients of degree t can be written