Dunkl Operators and Related Special Functions

TL;DR

This paper explores the harmonic analysis framework of Dunkl operators, which generalize classical special functions and polynomials through finite reflection groups and parameterized operator algebras.

Contribution

It introduces a harmonic analysis setting for Dunkl operators, connecting special functions with reflection groups and parameter-dependent operator algebras.

Findings

Dunkl operators generalize classical differential operators.

Special functions are linked to reflection groups and parameters.

The framework unifies various families of special functions.

Abstract

Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of functions like Gegenbauer, Jacobi and Jack symmetric polynomials and Bessel functions are labeled by parameters. These could be unspecified transcendental numbers or drawn from large sets of real numbers, for example the complement of {-1/2, -3/2, -5/2,...}. One aim of this chapter is to provide a harmonic analysis setting in which parameters play a natural role. The basic objects are finite reflection (Coxeter) groups and algebras of operators on polynomials which generalize the algebra of partial differential operators. These algebras have as many parameters as the number of conjugacy classes of reflections in the associated groups.