Sonine Formulas and Intertwining Operators in Dunkl Theory

Abstract

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 studybetween Dunkl operators with multiplicities k and k ′ . It has been a long-standing conjecture that V k ′ ,k is positive if k ′ ≥ k ≥ 0. We disprove this conjecture by constructing counterexamples for root system Bn. This matter is closely related to the existence of Soninetype 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.

Riesz distributions and Laplace transform in the Dunkl setting of type A

“…The proofs of Theorems 5.11 and 5.15 are based on analysis for multivariable hypergeometric functions which are given in terms of Jack polynomial expansions in the sense of [Kan,M3], combined with methods of Sokal [So] and a variant of the Shanbhag principle from [CL]. We finally mention that the generalized Wallach set (1.4) plays also an interesting role in connection with integral representations of Sonine type between Bessel functions for root system B n and the positivity of intertwining operators in the B n case, see [RV3].…”

Fractional Calculus in Russia at the End of XIX Century

“…then Formula (29) coincides with the definition of the fractional derivative given by Letnikov (28). In Reference [9], Sonine concluded that his formula cannot coincide with Grünwald-Letnikov formula for α > 0 without adding auxiliary function.…”

“…This formula was analyzed by Letnikov in his answer [7] on the remarks by Sonine. Letnikov proved that, under assumption that the function f is (m + 1) = ([α] + 1)-times continuously differentiable inside the circle γ, Formula (29) coincides with Letnikov's Formula (28). Note that Letnikov discussed Sonine's Formula (29) under the stronger assumptions on the function f .…”

“…Since it was shown that by integration by parts Formula ( 29) is reduced to the definition of the fractional derivative (28) without any auxiliary function, then Letnikov concluded that the following alternative holds: either (1) there exists no auxiliary function of the above form, or (2) formula ( 29) cannot be taken as the general definition of fractional derivative of an arbitrary order. Instead, he said that his definition is free of necessity to add an auxiliary function (in spite of the fact that this definition is reduced to different a form than ( 27) and ( 28) whenever α is negative or positive, respectively).…”

“…where Reν, Reµ > −1. Sonine formulas find interest in different questions of analysis (e.g., in Dunkl theory, as in Reference [28], or in the study of Levy processes [29]).…”