Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement

Abstract

Two families (type $A$ and type $B$) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type $A$) and the Laguerre polynomials (type $B$), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl.