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


SIGMA 4 (2008), 075, 7 pages      arXiv:0811.0507      http://dx.doi.org/10.3842/SIGMA.2008.075
Contribution to the Special Issue on Dunkl Operators and Related Topics

Generalized Bessel function of Type D

Nizar Demni
SFB 701, Fakultät für Mathematik, Universität Bielefeld, Deutschland

Received July 01, 2008, in final form October 24, 2008; Published online November 04, 2008

Abstract
We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D.

Key words: radial Dunkl processes; Brownian motions in Weyl chambers; generalized Bessel function; multivariate hypergeometric series.

pdf (207 kb)   ps (159 kb)   tex (10 kb)

References

  1. Baker T.H., Forrester P.J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216, solv-int/9608004.
  2. Beerends R.J., Opdam E.M., Certain hypergeometric series related to the root system BC, Trans. Amer. Math. Soc. 339 (1993), 581-607.
  3. Chybiryakov O., Skew-product representations of multidimensional Dunkl-Markov processes, Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), 593-611, arXiv:0808.3033.
  4. Dunkl C.F., Intertwining operators associated to the group S3, Trans. Amer. Math. Soc. 347 (1995), 3347-3374.
  5. Gallardo L., Yor M., Some new examples of Markov processes which enjoy the time-inversion property, Probab. Theory Related Fields 132 (2005), 150-162.
  6. Gallardo L., Yor M., A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab. 34 (2006), 1530-1549, math.PR/0609679.
  7. Gallardo L., Yor M., Some remarkable properties of the Dunkl martingales, in Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, Lecture Notes in Math., Vol. 1874, Springer, Berlin, 2006, 337-356.
  8. Gallardo L., Godefroy L., An invariance principle related to a process which generalizes N-dimensional Brownian motion, C. R. Math. Acad. Sci. Paris 338 (2004), 487-492.
  9. Grabiner D.J., Brownian motion in a Weyl chamber, non-colliding particles and random matrices, Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), 177-204, math.RT/9708207.
  10. Gross K.I., Richards D.St.P., Total positivity, spherical series and hypergeometric functions of matrix argument, J. Approx. Theor. 59 (1989), 224-246.
  11. Humphreys J.E., Reflections groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
  12. Kaneko J., Selberg integrals and hypergeometric functions with Jack polynomials, SIAM J. Math. Anal. 24 (1993), 1086-1110.
  13. Lebedev N.N., Special functions and their applications, Dover Publications, Inc., New York, 1972.
  14. Mcdonald L.G., Symmetric functions and Hall polynomials, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995.
  15. Muirhead R.J., Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982.
  16. Revuz D., Yor M., Continuous martingales and Brownian motion, 3rd ed., Springer-Verlag, Berlin, 1999.
  17. Rösler M., Voit M., Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575-643.
  18. Rösler M., Dunkl operator: 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.
  19. Schapira B., The Heckman-Opdam Markov processes, Probab. Theory Related Fields 138 (2007), 495-519.

Previous article   Next article   Contents of Volume 4 (2008)