Multivariate circular Jacobi polynomials

Abstract

We introduce a new multivariate orthogonal polynomial which is a 2-parameter deformation of the spherical polynomial by harmonic analysis on symmetric cone. This is also regarded as a multivariate analogue of the circular Jacobi polynomial. Further, the weight function of its orthogonality relation coincides with the circular Jacobi ensemble defined by Bourgade et al.. We also obtain its main properties : generating function, pseudo differential equation and deteraminant expression.