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

SIGMA 4 (2008), 083, 9 pages      arXiv:0812.0739      https://doi.org/10.3842/SIGMA.2008.083
Contribution to the Special Issue on Dunkl Operators and Related Topics

A Limit Relation for Dunkl-Bessel Functions of Type A and B

Margit Rösler a and Michael Voit b
a) Institut für Mathematik, TU Clausthal, Erzstr. 1, D-38678 Clausthal-Zellerfeld, Germany
b) Fachbereich Mathematik, TU Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany

Received October 21, 2008, in final form November 26, 2008; Published online December 03, 2008

Abstract
We prove a limit relation for the Dunkl-Bessel function of type BN with multiplicity parameters k1 on the roots ±ei and k2 on ±ei±ej where k1 tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type AN−1 with multiplicity k2. For certain values of k2 an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones.

Key words: Bessel functions; Dunkl operators; asymptotics.

pdf (241 kb)   ps (170 kb)   tex (12 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. Baker T.H., Forrester P.J., Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95 (1998), 1-50, q-alg/9612003.
  3. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  4. Faraut J., Korányi A., Analysis on symmetric cones, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.
  5. Gross K., Richards D., Special functions of matrix argument I. Algebraic induction, zonal polynomials, and hypergeometric functions, Trans. Amer. Math. Soc. 301 (1987), 781-811.
  6. Helgason S., Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Pure and Applied Mathematics, Vol. 113, Academic Press, Inc., Orlando, FL, 1984.
  7. Herz C.S., Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474-523.
  8. Kaneko J., Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993), 1086-1100.
  9. Knop F., Sahi S., A recursion and combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 9-22, q-alg/9610016.
  10. Opdam E.M., Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), 333-373.
  11. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Editors E. Koelink et al., Springer Lect. Notes Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.
  12. Rösler M., A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), 2413-2438, math.CA/0210137.
  13. Rösler M., Bessel convolutions on matrix cones, Compos. Math. 143 (2007), 749-779, math.CA/0512474.
  14. Rösler M., Voit M., Limit theorems for radial random walks on p×q matrices as p tends to infinity, Math. Nachr., to appear, math.CA/0703520.
  15. Stanley R.P., Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76-115.
  16. Voit M., A limit theorem for isotropic random walks on Rd for d®¥, Russian J. Math. Phys. 3 (1995), 535-539.
  17. Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1966.

Previous article   Next article   Contents of Volume 4 (2008)