Limits of BC-type orthogonal polynomials as the number of variables goes to infinity

TL;DR

This paper investigates the asymptotic behavior of BC-type Jacobi polynomials as the number of variables increases, revealing how they approximate spherical functions of infinite-dimensional symmetric spaces.

Contribution

It extends previous work on Jack polynomials by analyzing BC-type Jacobi polynomials' limits and their relation to infinite-dimensional symmetric spaces.

Findings

Asymptotic behavior of BC-type Jacobi polynomials characterized

Approximation of infinite-dimensional spherical functions established

Connections to symmetric spaces of types B, C, D, BC clarified

Abstract

We describe the asymptotic behavior of the multivariate BC-type Jacobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results describe the approximation of the spherical functions of the infinite-dimensional symmetric spaces of type B,C,D or BC by the spherical functions of the corresponding finite-dimensional symmetric spaces. Similar results for the Jack polynomials were established in our earlier paper (Intern. Math. Res. Notices 1998, no. 13, 641-s682; arXiv:q-alg/9709011). The main results of the present paper were obtained in 1997.