Generalized Ellipsoidal and Sphero-Conal Harmonics

Abstract

Abstract. Classical ellipsoidal and sphero-conal harmonics are polynomial solutions of the Laplace equation that can be expressed in terms of Lamé polynomials. Generalized ellipsoidal and sphero-conal harmonics are polynomial solutions of the more general Dunkl equation that can be expressed in terms of Stieltjes polynomials. Niven's formula connecting ellipsoidal and sphero-conal harmonics is generalized. Moreover, generalized ellipsoidal harmonics are applied to solve the Dirichlet problem for Dunkl's equation on ellipsoids.

Reflection groups in analysis and applications

“…The polynomials even in each x i and restricted to S N−1 can be considered as polynomials in y 1 ,... ,y N−1 where y i := x 2 i and y N : Dai and Xu [6] showed there is a critical exponent for Cèsaro summability, namely N 2 + γ − 1 − min 1≤i≤N κ i . Volkmer [52] considered Dirichlet problems for Δ κ in ellipsoids, by using sphero-conal harmonics, expressed as Stieltjes quasi-polynomials.…”

External Ellipsoidal Harmonics for the Dunkl-Laplacian

“…The function Φ(·, z) is analytic and hharmonic on and inside J. Therefore, it can be expanded in ellipsoidal h-harmonics as in [21,Section 7] and the expansion coefficients can be evaluated using (4.5). This gives (4.9) after a simple calculation.…”

“…which holds for any h-harmonic function g; see the formula before (6.3) in [21]. Then we replace the operator −Λ by a 0 D 2 0 + · · · + a k D 2 k which is possible because Φ(·, z) is h-harmonic.…”

“…Therefore, the Stieltjes polynomial E n,0 is a multiple of P (α,β) n . In [21] the Stieltjes polynomials were normalized such that their leading coefficient equals 1. This gives E n,0 = a n P (α,β) n , where a n = a (α,β) n := n!2 n (α + β + n + 1) n .…”

Hobson's formula for Dunkl operators and its applications

“…The Dunkl operators are differentialdifference operator on a Euclidean space R d associated with a finite reflection group G, which are deformations of directional derivatives. For G = Z d 2 , our formula is previously given by Volkmer [30]. Moreover, our formula contains the original Hobson's formula as a special case.…”