Formulas for elementary spherical functions and generalized Jacobi polynomials. (English) Zbl 0549.43006

Vretare, Lars

SIAM J. Math. Anal. 15, 805-833 (1984).

Generalized Jacobi polynomials are symmetric polynomials \(p^{\alpha,\beta,\gamma}_{n_ 1,...,n_{\ell}}(x_ 1,...,x_{\ell})\) which are orthogonal with respect to the weight function \(\prod_{1\leq i\leq\ell }(1-x_ i)^{\alpha}(1+x_ i)^{\beta}\prod_{i<j}(x_ i-x_ j)^{2\gamma +1},\quad -1\leq x_{\ell}\leq x_{\ell -1}\leq...\leq x_ 1\leq 1.\) For certain values of the parameters \(\alpha\), \(\beta\), \(\gamma\) the polynomials are interpreted as elementary spherical functions on symmetric spaces. A number of formulas are proved, in particular in two and three variables.

MSC:

43A90	Harmonic analysis and spherical functions
22E46	Semisimple Lie groups and their representations
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

root systems; c-function of Harish Chandra; recurrence formulas; polynomials; spherical functions; symmetric spaces