Jacobi Polynomials as Spherical Harmonics

Symmetric Diffusions with Polynomial Eigenvectors

“…In the same way, as we shall see below in Section 4, when a = b = (d − 1)/2, J a,a may be seen as the Laplace operator ∆ S d−1 on the unit sphere S d−1 in R d acting on functions depending only on the first coordinate (or equivalently on functions invariant under the rotations leaving (1, 0, · · · , 0) invariant), and a similar interpretation is valid for J p/2,q/2 for integers p and q. This interpretation comes from Zernike and Brinkman [8] and Braaksma and Meulenbeld [6] (see also Koornwinder [15]). Jacobi polynomials also play a central role in the analysis on compact Lie groups.…”

On pulsating strings in Schrodinger backgrounds

“…We can further transform the Jacobi polynomial to the standard spherical harmonics P 2n (cos θ), i.e. P [47], where m = 2n should be even. Thus the final polynomial solution of Eq.…”

Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres

“…It turns out that explicit formulae for the orthogonal polynomials can be given in terms of orthogonal polynomials with respect to the weight function (1 x 2 ) ï 1Û2 jxj 2ñ on [ 1,1]. These polynomials of one variable can be written explicitly in terms of Jacobi polynomials, but they seem to possess properties that make them closer to the Gegenbauer polynomials; we shall call them generalized Gegenbauer polynomials.…”

“…2 ) is chosen so that the integral of w (ï,ñ) over the integral [ 1,1] is 1. We denote the orthonormal polynomials with respect to the weight function w (ï,ñ) by D (ï,ñ) n ; that is,…”

“…Thus, the intertwining operator shows that the h-harmonics can be written as integrals against trigonometric functions. In fact, the analytic proof of the formula (2.3) is based on the following formula in [1],…”