A unified approach to hypergeometric class functions

TL;DR

This paper introduces a unified framework for hypergeometric class functions using complex Lie algebras, providing new insights into their properties and interrelations through algebraic and differential operator techniques.

Contribution

It presents a novel unified approach to hypergeometric functions based on Lie algebra representations, encompassing recurrence, symmetry, and orthogonality properties.

Findings

Unified treatment of hypergeometric functions and their properties.

Representation of hypergeometric functions as eigenfunctions of Lie algebra Casimir operators.

Derivation of recurrence relations, symmetries, and orthogonality within a single framework.

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.