A review of multivariate orthogonal polynomials

Abstract

This paper contains a brief review of orthogonal polynomials in two and several variables. It supplements the Koornwinder survey [40]. Several recently discovered systems of orthogonal polynomials have been treated in this work. We did not provide any proofs of the theorem presented here but references to the original sources are given for the benefit of the interested reader. It is hoped that collecting these scattered results in one place will make them accessible for the user.

Keywords

• Disc polynomials
• Zernike polynomials
• 2D-Laguerre polynomials
• q-2D-Laguerre polynomials
• Generating functions
• Ladder operators
• q-Sturm–Liouville equations
• q-integrals
• q-Zernike polynomials
• 2D-Jacobi polynomials
• q-2D-Jacobi polynomials
• Connection relations
• Biorthogonal functions
• Generating functions
• Rodrigues formulas
• Zeros