# Sonine formulas and intertwining operators in Dunkl theory

## TL;DR

This paper disproves a long-standing conjecture by showing that certain Dunkl intertwining operators are not always positive, revealing new limitations in the structure of Dunkl theory and related special functions.

## Contribution

It constructs explicit counterexamples for root system B_n, demonstrating that the generalized Dunkl intertwining operators are not necessarily positive when multiplicities increase.

## Key findings

- Counterexamples for root system B_n where $V_{k',k}$ is not positive
- Existence of cases where Sonine formulas between Dunkl kernels and Bessel functions fail
- Necessary conditions for Sonine-type integral formulas and positivity in multivariable special functions

## Abstract

Let $V_k$ denote Dunkl's intertwining operator associated with some root system $R$ and multiplicity function $k$. For two multiplicities $k, k^\prime$ on $R$, we study the operator $V_{k^\prime,k} = V_{k^\prime}\circ V_k^{-1}$, which intertwines the Dunkl operators for multiplicity $k$ with those for multiplicity $k^\prime.$ While it is well-known that the operator $V_k$ is positive for nonnegative $k$, it has been a long-standing conjecture that its generalizations $V_{k^\prime,k}$ are also positive if $k^\prime \geq k \geq 0,$ which is known to be true in rank one. In this paper, we disprove this conjecture by constructing examples for root system $B_n$ with multiplicites $k^\prime \geq k \geq 0$ for which $V_{k^\prime, k}$ is not positive. This matter is closely related to the existence of integral representations of Sonine type between the Dunkl kernels and Bessel functions associated with the relevant multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine-type integral formulas for Heckman-Opdam hypergeometric functions of type $BC_n$ as well as conditions on the existence of positive branching coefficients between systems of multivariable Jacobi polynomials.