SIGMA 4 (2008), 091, 13 pages      arXiv:0812.4365      https://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 P_n^α,β and Jacobi's functions Q_n^α,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.

pdf (268 kb)   ps (181 kb)   tex (15 kb)

References

1. Arscott F.M., Periodic differential equations. An introduction to Mathieu, Lamé, and allied functions, International Series of Monographs in Pure and Applied Mathematics, Vol. 66, A Pergamon Press Book, The Macmillan Co., New York, 1964.
2. Ben Saïd S., Ørsted B., The wave equation for Dunkl operators, Indag. Math. (N.S.) 16 (2005), 351-391.
3. Blimke J., Myklebust J., Volkmer H., Merrill S., Four-shell ellipsoidal model employing multipole expansion in ellipsoidal coordinates, Med. Biol. Eng. Comput. 46 (2008), 859-869.
4. Dassios G., The magnetic potential for the ellipsoidal MEG problem, J. Comput. Math. 25 (2007), 145-156.
5. Dassios G., Kariotou F., Magnetoencephalography in ellipsoidal geometry, J. Math. Phys. 44 (2003), 220-241.
6. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
7. Dunkl C.F., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
8. Dunkl C.F., A Laguerre polynomial orthogonality and the hydrogen atom, Anal. Appl. (Singap.) 1 (2003), 177-188, math-ph/0011021.
9. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
10. Evans L., Partial differential equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998.
11. Heine E., Handbuch der Kugelfunktionen, Vol. 1, G. Reimer Verlag, Berlin, 1878.
12. Heine E., Handbuch der Kugelfunktionen, Vol. 2, G. Reimer Verlag, Berlin, 1881.
13. Hikami K., Boundary K-matrix, elliptic Dunkl operator and quantum many-body systems, J. Phys. A: Math. Gen. 29 (1996), 2135-2147.
14. Hobson E.W., The theory of spherical and ellipsoidal harmonics, Cambridge University Press, Cambridge, 1931.
15. Kalnins E.G., Miller W. Jr., Jacobi elliptic coordinates, functions of Heun and Lamé type and the Niven transform, Regul. Chaotic Dyn. 10 (2005), 487-508.
16. Niven W.D., On ellipsoidal harmonics, Phil. Trans. Royal Society London A 182 (1891), 231-278.
17. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.
18. Rösler M., Voit M., Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575-643.
19. Stieltjes T.J., Sur certains polynômes qui vérifient une équation différentielle linéaire du second ordre et sur la théorie des fonctions de Lamé, Acta Math. 5 (1885), 321-326.
20. Szegö G., Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975.
21. Volkmer H., Generalized ellipsoidal and sphero-conal harmonics, SIGMA 2 (2006), 071, 16 pages, math.CA/0610718.
22. Whittaker E.T., Watson G.N., A course in modern analysis, Cambridge University Press, Cambridge, 1927.
23. Xu Y., Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), 175-192.