Jack polynomials, generalized binomial coefficients and polynomial solutions of the generalized Laplace's equation

Abstract

We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace's equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials and the constraints on the coefficients of expansion are derived. These constraints involve generalized binomial coefficients defined through Jack polynomials. Generalized binomial coefficients for partitions of $k$ upto $k=6$ are tabulated.