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SIGMA 4 (2008), 075, 7 pages arXiv:0811.0507
https://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.
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References
- Baker T.H., Forrester P.J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216, solv-int/9608004.
- Beerends R.J., Opdam E.M., Certain hypergeometric series related to the root system BC, Trans. Amer. Math. Soc. 339 (1993), 581-607.
- Chybiryakov O., Skew-product representations of multidimensional Dunkl-Markov processes, Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), 593-611, arXiv:0808.3033.
- Dunkl C.F., Intertwining operators associated to the group S3, Trans. Amer. Math. Soc. 347 (1995), 3347-3374.
- Gallardo L., Yor M., Some new examples of Markov processes which enjoy the time-inversion property, Probab. Theory Related Fields 132 (2005), 150-162.
- Gallardo L., Yor M., A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab. 34 (2006), 1530-1549, math.PR/0609679.
- 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.
- 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.
- 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.
- Gross K.I., Richards D.St.P., Total positivity, spherical series and hypergeometric functions of matrix argument, J. Approx. Theor. 59 (1989), 224-246.
- Humphreys J.E., Reflections groups and Coxeter groups,
Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
- Kaneko J., Selberg integrals and hypergeometric functions with Jack polynomials, SIAM J. Math. Anal. 24 (1993), 1086-1110.
- Lebedev N.N., Special functions and their applications, Dover Publications, Inc., New York, 1972.
- Mcdonald L.G., Symmetric functions and Hall polynomials, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995.
- Muirhead R.J., Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics,
John Wiley & Sons, Inc., New York, 1982.
- Revuz D., Yor M., Continuous martingales and Brownian motion, 3rd ed., Springer-Verlag, Berlin, 1999.
- Rösler M., Voit M., Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575-643.
- 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.
- Schapira B., The Heckman-Opdam Markov processes, Probab. Theory Related Fields 138 (2007), 495-519.
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