Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 071, 16 pages      math.CA/0610718      http://dx.doi.org/10.3842/SIGMA.2006.071
Contribution to the Vadim Kuznetsov Memorial Issue

Generalized Ellipsoidal and Sphero-Conal Harmonics

Hans Volkmer
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201 USA

Received August 25, 2006, in final form October 20, 2006; Published online October 24, 2006

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.

Key words: generalized ellipsoidal harmonic; Stieltjes polynomials; Dunkl equation; Niven formula.

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