Sonine Formulas and Intertwining Operators in Dunkl Theory

Let V-k denote Dunkl's intertwining operator associated with some root system R and multiplicity k. For two multiplicities k, k' on R, we study the intertwiner V-k',V-k = V-k' omicron V-k(-1) between Dunkl operators with multiplicities k and k'. It has been a long-standing conjecture that V-k',V-k is positive if k' >= k >= 0. We disprove this conjecture by constructing counterexamples for root system B-n. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman-Opdam hypergeometric functions of type BCn and conditions for positive branching coefficients between multivariable Jacobi polynomials.