Formulas for Elementary Spherical Functions and Generalized Jacobi Polynomials

Abstract

Elementary spherical functions on symmetric spaces can be considered as orthogonal polynomials in several variables. This paper deals with the weight function \[ w\left( {x_1 , \cdots ,x_l } \right) = \Pi \left( {1 - x_i } \right)^\alpha \left( {1 + x_i } \right)^\beta \Pi \left( {x_i - x_j } \right)^{2\gamma + 1} ,\quad - 1 \leq x_i \leq 1.\] Recurrence relations for polynomials corresponding to different spaces are derived and generalized to other values of the parameters $\alpha $, $\beta $ and $\gamma $.