ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS

Abstract

The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of A₁ and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of A₂ and a specific version of two variable Jacobi polynomials. The connection with recently studied G₂-polynomials is established. Formulas for connection between the four types of orbit functions of B_n or C_n and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.