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 $k_1\geq 0, k_2>0$. 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 $h\in [k_2(n-1),\infty [ \cup (\{0,k_2,\ldots ,k_2(n-1)\}-\mathbb Z_+)$. 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 $k_1 \geq 0, k_2\in \{1/2,1,2\}$ 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 $k_1\geq 0, k_2>0$ 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.