The Ramanujan Journal

, Volume 34, Issue 1, pp 39–55 | Cite as

Special functions associated with complex reflection groups

Article
First Online:
Received:
Accepted:

Abstract

In this paper, we first review the theory of Dunkl operators for complex reflection groups and then the theory of hyper-Bessel functions, which are a particular case of Meijer’s G-function and satisfy a higher order differential equation. Then we show that there exists a close relation between both theories. In fact, the components of the eigenfunctions of a Dunkl operator for a complex reflection group in the rank one case can be expressed in terms of hyper-Bessel functions.

Keywords

Special functions Meijer’s G-function 

Mathematics Subject Classification (2000)

33E30 33C52 

References

  1. 1.
    Ben Cheikh, Y.: Differential equations satisfied by the components with respect to the cyclic group of order n of some special functions. J. Math. Anal. Appl. 244(2), 483–497 (2000) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Davis, P.J.: Circulant Matrices, 2nd edn. Chelsea, New York (1994) MATHGoogle Scholar
  3. 3.
    Delerue, P.: Sur le calcul symbolique à n variables et les fonctions hyperbesséliennes. II. Fonctions hyperbesséliennes. Ann. Soc. Sci. Brux., Sér. I 67, 229–274 (1953) MATHMathSciNetGoogle Scholar
  4. 4.
    Dimovski, I.H., Kiryakova, V.S.: Generalized Poisson transmutations and corresponding representations of hyper-Bessel functions. C. R. Acad. Bulgare Sci. 39(10), 29–32 (1986) MATHMathSciNetGoogle Scholar
  5. 5.
    Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dunkl, C.F., Opdam, E.M.: Dunkl operators for complex reflection groups. Proc. Lond. Math. Soc. 86(3), 70–108 (2003) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Vol. III. McGraw-Hill, New York (1955) MATHGoogle Scholar
  8. 8.
    Fitouhi, A., Dhaouadi, L., Bouzeffour, F.: r-Extension of Dunkl operator in one variable and Bessel functions of vector index (2013). arXiv:1209.5277v3 [math.FA]
  9. 9.
    Kiryakova, V.S.: New integral representations of the generalized hypergeometric functions. C. R. Acad. Bulgare Sci. 39(12), 33–36 (1986) MATHMathSciNetGoogle Scholar
  10. 10.
    Kiryakova, V.S.: Obrechkoff integral transform and hyper-Bessel operators via G-function and fractional calculus approach. Glob. J. Pure Appl. Math. 1, 321–341 (2005) MATHMathSciNetGoogle Scholar
  11. 11.
    Klyuchantsev, M.I.: Singular differential operators with r−1 parameters and Bessel functions of a vector index. Sib. Math. J. 24, 353–367 (1983) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Muldoon, M.E.: Generalized hyperbolic functions, circulant matrices and functional equations. Linear Algebra Appl. 406, 272–284 (2005) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia

Personalised recommendations