Hypergeometric Functions and Their Generalizations to Higher Dimensions: A study of the classical hypergeometric function and its generalizations associated with root systems

Abstract

The hypergeometric differential equation is a classical ODE of second order, and it was already studied by Gauss. The hypergeometric function is classically defined as the solution to this equation that is analytic at x = 0. With this definition it is not obvious how to generalize the hypergeometric function to higher dimensions. With a shift in perspective we can arrive at the same differential equation by studying a certain eigenvalue problem of polynomials of so called Dunkl operators. These are easier to generalize and will lead us to hypergeometric functions associated with so called root systems in higher dimension.