Geodesic
A Limit Relation for Dunkl–Bessel Functions of Type A and B
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove a limit relation for the Dunkl–Bessel function of type $B_N$ with multiplicity parameters $k_1$ on the roots $\pm e_i$ and $k_2$ on $\pm e_i\pm e_j$ where $k_1$ tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type $A_{N-1}$ with multiplicity $k_2$. For certain values of $k_2$ an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones.
Keywords: Bessel functions; Dunkl operators; asymptotics. 
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     author = {Margit R\"osler and Michael Voit},
     title = {A~Limit {Relation} for {Dunkl{\textendash}Bessel} {Functions} of {Type~A} {and~B}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a82/}
}
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Margit Rösler; Michael Voit. A Limit Relation for Dunkl–Bessel Functions of Type A and B. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a82/

[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 | DOI | MR | Zbl

[2] Baker T. H., Forrester P. J., “Nonsymmetric Jack polynomials and integral kernels”, Duke Math. J., 95 (1998), 1–50 ; q-alg/9612003 | DOI | MR | Zbl

[3] Dunkl C. F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, 81, Cambridge University Press, Cambridge, 2001 | MR | Zbl

[4] Faraut J., Korányi A., Analysis on symmetric cones, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994 | MR | Zbl

[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 | DOI | MR | Zbl

[6] Helgason S., Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Pure and Applied Mathematics, 113, Academic Press, Inc., Orlando, FL, 1984 | MR | Zbl

[7] Herz C. S., “Bessel functions of matrix argument”, Ann. of Math. (2), 61 (1955), 474–523 | DOI | MR | Zbl

[8] Kaneko J., “Selberg integrals and hypergeometric functions associated with Jack polynomials”, SIAM J. Math. Anal., 24 (1993), 1086–1100 | DOI | MR

[9] Knop F., Sahi S., “A recursion and combinatorial formula for Jack polynomials”, Invent. Math., 128 (1997), 9–22 ; q-alg/9610016 | DOI | MR | Zbl

[10] Opdam E. M., “Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group”, Compositio Math., 85 (1993), 333–373 | MR | Zbl

[11] Rösler M., “Dunkl operators: theory and applications”, Orthogonal Polynomials and Special Functions (Leuven, 2002), Springer Lect. Notes Math., 1817, eds. E. Koelink et al., Springer, Berlin, 2003, 93–135 ; math.CA/0210366 | MR | Zbl

[12] Rösler M., “A positive radial product formula for the Dunkl kernel”, Trans. Amer. Math. Soc., 355 (2003), 2413–2438 ; math.CA/0210137 | DOI | MR | Zbl

[13] Rösler M., “Bessel convolutions on matrix cones”, Compos. Math., 143 (2007), 749–779 ; math.CA/0512474 | MR | Zbl

[14] Rösler M., Voit M., Limit theorems for radial random walks on $p\times q$ matrices as $p$ tends to infinity, Math. Nachr. to appear | MR

[15] Stanley R. P., “Some combinatorial properties of Jack symmetric functions”, Adv. Math., 77 (1989), 76–115 | DOI | MR | Zbl

[16] Voit M., “A limit theorem for isotropic random walks on $\mathbb R^d$ for $d\to\infty$”, Russian J. Math. Phys., 3 (1995), 535–539 | MR | Zbl

[17] Watson G. N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1966 | MR