A matrix valued solution to Bochner?s problem

F.A. GRUNBAUM, I. PACHARONI, J. TIRAO

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL

IOP PUBLISHING LTD

Año: 2001 vol. 34 p. 10647 - 10656

1751-8113

Back in 1929, S. Bochner [B] posed the problem of determining all families of scalar valued orthogonal polynomials that are eigenfunctions of some arbitrary but fixed second order differential operator. Many of the topics discussed in these papers have made interesting contacts with areas as varied as integrable systems, random matrix theory, interpolation and approximation theory, problems of electrostatic equilibrium, extensions of the Huygens? principle, representations of the Weyl algebra,Calogero-Moser systems, etc. It appears very natural, and maybe even profitable, to revisit the question in the matrix valued context and this is the theme of this paper. Returning to the question raised by Bochner it is clear, starting with the work that stretches from Cartan to Harish-Chandra, see [GV], that a natural place to find examples satisfying his conditions is in the theory of spherical functions for symmetric spaces of rank one. In the compact case this leads to Jacobi polynomials, one of the four families that feature in the full solution of Bochner?s problem.