Positive intertwiners for Bessel functions of type B

TL;DR

This paper investigates conditions under which Dunkl intertwining operators for root system B are positive, establishing new partial converse results that connect operator positivity with positive Sonine formulas for Bessel functions.

Contribution

It provides new partial converse positivity results for Dunkl operators of type B, linking operator positivity to positive Sonine formulas and Bessel functions.

Findings

Positivity of certain Dunkl operators is established for specific parameter ranges.

Positive Sonine formulas are shown to exist under these positivity conditions.

The proofs utilize Laguerre polynomial connections and Bessel function approximations.

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.