Fourier transforms of orthogonal polynomials on the cone

TL;DR

This paper derives Fourier transforms for multivariate orthogonal polynomials on the cone, introduces new orthogonal functions, and connects results to continuous Hahn polynomials, advancing mathematical understanding of these functions.

Contribution

It provides explicit Fourier transforms for cone-based orthogonal polynomials and defines new multivariate orthogonal functions using Parseval's identity.

Findings

Fourier transforms of Laguerre and Jacobi polynomials on the cone are obtained.

New families of multivariate orthogonal functions are introduced.

Results are expressed in terms of continuous Hahn polynomials.

Abstract

The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal polynomials on the cone such as Laguerre polynomials on the cone and Jacobi polynomials on the cone and to define two new families of multivariate orthogonal functions by using Parseval's identity. Also, the obtained results are expressed in terms of the continuous Hahn polynomials.