Positive intertwiners for Bessel functions of type B

Abstract

Let V k V_k denote Dunkl’s intertwining operator for the root sytem B n B_n with multiplicity k = ( k 1 , k 2 ) k=(k_1,k_2) with k 1 ≥ 0 , k 2 > 0 k_1\geq 0, k_2>0 . It was recently shown that the positivity of the operator V k ′ , k = V k ′ ∘ V k − 1 V_{k^\prime \!,k} = V_{k^\prime }\circ V_k^{-1} which intertwines the Dunkl operators associated with k k and k ′ = ( k 1 + h , k 2 ) k^\prime =(k_1+h,k_2) implies that h ∈ [ k 2 ( n − 1 ) , ∞ [ ∪ ( { 0 , k 2 , … , k 2 ( n − 1 ) } − Z + ) 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 ≥ 0 , k 2 ∈ { 1 / 2 , 1 , 2 } k_1 \geq 0, \,k_2\in \{1/2,1,2\} and h > k 2 ( n − 1 ) h>k_2(n-1) , the operator V k ′ , k 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 B_n . Moreover, the same positivity results hold for arbitrary k 1 ≥ 0 , k 2 > 0 k_1\geq 0, k_2>0 and h ∈ k 2 ⋅ Z + . 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.