Fourier Transforms of Orthogonal Structures on the Paraboloid

Abstract The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal structures on the paraboloid such as Laguerre polynomials on the paraboloid and Jacobi polynomials on the paraboloid, and to define two new families of multivariate orthogonal functions by using Parseval’s identity. In addition, some contiguous relations for these families of functions are given and the obtained results are expressed in terms of the continuous Hahn polynomials. Finally, we provide graphical illustrations of some special functions and their Fourier transforms.