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

SIGMA 4 (2008), 093, 11 pages      arXiv:0812.4819
Contribution to the Special Issue on Dunkl Operators and Related Topics

An Alternative Definition of the Hermite Polynomials Related to the Dunkl Laplacian

Hendrik De Bie
Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Krijgslaan 281, 9000 Gent, Belgium

Received October 07, 2008, in final form December 18, 2008; Published online December 28, 2008

We introduce the so-called Clifford-Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related to the Dunkl operators (see [Rösler M., Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.]) as well as with the basis of the weighted L2 space introduced by Dunkl.

Key words: Hermite polynomials; Dunkl operators; Clifford analysis.

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  1. Ben Saïd S., On the integrability of a representation of sl(2,R), J. Funct. Anal. 250 (2007), 249-264.
  2. Ben Saïd S., Ørsted B., Segal-Bargmann transforms associated with finite Coxeter groups, Math. Ann. 334 (2006), 281-323.
  3. Brackx F., Delanghe R., Sommen F., Clifford analysis, Research Notes in Mathematics, Vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982.
  4. Brackx F., de Schepper N., Sommen F., The higher dimensional Hermite transform: a new approach, Complex Var. Theory Appl. 48 (2003), 189-210.
  5. Brackx F., Sommen F., Clifford-Hermite wavelets in Euclidean space, J. Fourier Anal. Appl. 6 (2000), 299-310.
  6. Cerejeiras P., Kähler U., Ren G., Clifford analysis for finite reflection groups, Complex Var. Elliptic Equ. 51 (2006), 487-495.
  7. De Bie H., Fourier transform and related integral transforms in superspace, J. Math. Anal. Appl. 345 (2008), 147-164, arXiv:0805.1918.
  8. De Bie H., Sommen F., Hermite and Gegenbauer polynomials in superspace using Clifford analysis, J. Phys. A: Math. Theor. 40 (2007), 10441-10456, arXiv:0707.2863.
  9. De Bie H., Sommen F., Spherical harmonics and integration in superspace, J. Phys. A: Math. Theor. 40 (2007), 7193-7212, arXiv:0705.3148.
  10. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
  11. Delanghe R., Sommen F., Soucek V., Clifford algebra and spinor-valued functions, Mathematics and Its Applications, Vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992.
  12. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  13. Dunkl C.F., Hankel transforms associated to finite reflection groups, in Proc. of the Special Session on Hypergeometric Functions on Domains of Positivity, Jack polynomials and Applications (Tampa 1991), Contemp. Math. 138 (1992), 123-138.
  14. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  15. Fueter R., Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen, Comment. Math. Helv. 7 (1934), 307-330.
  16. Gilbert J.E., Murray M.A.M., Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, Vol. 26, Cambridge University Press, Cambridge, 1991.
  17. Heckman G.J., A remark on the Dunkl differential-difference operators, in Harmonic Analysis on Reductive Groups (Brunswick, ME, 1989), Editors W. Barker and P. Sally, Progress in Math., Vol. 101, Birkhäuser Boston, Boston, MA, 1991, 181-191.
  18. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
  19. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  20. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
  21. Sommen F., Monogenic functions on surfaces, J. Reine Angew. Math. 361 (1985), 145-161.
  22. Sommen F., On a generalization of Fueter's theorem, Z. Anal. Anwendungen 19 (2000), 899-902.
  23. Sommen F., Special functions in Clifford analysis and axial symmetry, J. Math. Anal. Appl. 130 (1988), 110-133.
  24. Harmonic polynomials associated with reflection groups, Canad. Math. Bull. 43 (2000), 496-507.

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