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


SIGMA 4 (2008), 091, 13 pages      arXiv:0812.4365      http://dx.doi.org/10.3842/SIGMA.2008.091
Contribution to the Special Issue on Dunkl Operators and Related Topics

External Ellipsoidal Harmonics for the Dunkl-Laplacian

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

Received September 22, 2008, in final form December 18, 2008; Published online December 23, 2008

Abstract
The paper introduces external ellipsoidal and external sphero-conal h-harmonics for the Dunkl-Laplacian. These external h-harmonics admit integral representations, and they are connected by a formula of Niven's type. External h-harmonics in the plane are expressed in terms of Jacobi polynomials Pnα,β and Jacobi's functions Qnα,β of the second kind.

Key words: external ellipsoidal harmonics; Stieltjes polynomials; Dunkl-Laplacian; fundamental solution; Niven's formula; Jacobi's function of the second kind.

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