Positive intertwiners for Bessel functions of type B

Abstract

Let $V_k$ denote Dunkl's intertwining operator for the root sytem $B_n$ with multiplicity $k=(k_1,k_2)$ with $k_1\ge 0,k_2>0$. It was recently shown that the positivity of the operator $V_{k^{\prime },k}=V_{k^{\prime }}\circ V_k^{-1}$ which intertwines the Dunkl operators associated with $k$ and $k^{\prime }=(k_1+h,k_2)$ implies that $h\in [k_2(n-1),\infty [\cup (\{0,k_2,\dots ,k_2(n-1)\}-{\mathbb{Z}}_{+})$. This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: For $k_1\ge 0,k_2\in \{1/2,1,2\}$ and $h>k_2(n-1)$, the operator $V_{k^{\prime },k}$ is positive when restricted to functions which are invariant under the Weyl group, and there is an associated positive Sonine formula for the Bessel functions of type $B_n$. Moreover, the same positivity results hold for arbitrary $k_1\ge 0,k_2>0$ and $h\in k_2\cdot {\mathbb{Z}}_{+}.$ The proof is based on a formula of Baker and Forrester on connection coefficients between multivariate Laguerre polynomials and an approximation of Bessel functions by Laguerre polynomials.

13 pages