A unified approach to hypergeometric class functions

Abstract: Hypergeometric class equations are given by second order differential operators in one variable whose coefficient at the second derivative is a polynomial of degree $\leq2$, at the first derivative of degree $\leq1$ and the free term is a number. Their solutions, called hypergeometric class functions, include the Gauss hypergeometric function and its various limiting cases. The paper presents a unified approach to these functions. The main structure behind this approach is a family of complex 4-dimensional Lie algebras, originally due to Willard Miller. Hypergeometric class functions can be interpreted as eigenfunctions of the quadratic Casimir operator in a representation of Miller's Lie algebra given by differential operators in three complex variables. One obtains a unified treatment of various properties of hypergeometric class functions such as recurrence relations, discrete symmetries, power series expansions, integral representations, generating functions and orthogonality of polynomial solutions.