The Lie Theory of Two‐Variable Hypergeometric Functions

Abstract

We develop a Lie‐algebraic method that associates with each of the 34 distinct second‐order hypergeometric functions in two variables a canonical system of partial differential equations. The special functions arise by partial separation of variables in these simple systems. Some consequences are a demonstration that all such functions appear as solutions of the 4‐variable wave equation and a classification of the possible imbeddings. In each case the functions are characterized by first‐ and second‐order operators in the enveloping algebra of the conformal symmetry algebra for the wave equation. In some cases the 3‐variable wave and heat equations and the 2‐variable Helmholtz equation also arise. This intimate relationship between Horn functions and some fundamental equations of mathematical physics shows that these functions are more interesting than was previously recognized and permits use of the powerful tools of Lie theory and separation of variables to obtain properties of the functions.