We introduce and study a class of analytic difference operators admitting reflectionless eigenfunctions. Our construction of the class is patterned after the Inverse Scattering Transform for the reflectionless self-adjoint Schrödinger and Jacobi operators corresponding to KdV and Toda lattice solitons.

Abstract. We begin with the study of some properties of the radial Dunkl process associated to a reduced root system R. It is shown that this diffusion is the unique strong solution for all t ≥ 0 of a SDE with singular drift. Then, we study T0, the first hitting time of the positive Weyl chamber : we prove, via stochastic calculus, a result already obtained by Chybiryakov on the finiteness of T...

A singular polynomial is one which is annihilated by all Dunkl operators for a certain parameter value. These polynomials were first studied by Dunkl, de Jeu and Opdam, (Trans. Amer. Math. Soc. 346 (1994), 237-256). This paper constructs a family of such polynomials associated to the irreducible representation (N − 2, 1, 1) of the symmetric group SN for odd N and parameter values − 1 2 ,− 3 2 ,...

Abstract. We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the W -invariant Dunkl–Hermite polynomials. Illustrative examples are given by the irreducible r...

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on RN . The Dunkl operators are denoted by Tj for 1 ≤ j ≤ N, and the Laplacian ∆κ = ∑j=1 T2 j . This paper finds the homogeneous harmonic polynomials annihilated by all Tj for j > 2. The structure constants with respect to the Gaussian and sphere inner...

In this paper, we investigate the well-posedness for Cauchy problem multi-term time-fractional heat equation associated with Dunkl operator. The under consideration includes a linear combination of Caputo derivatives in time decreasing orders (0, 1) and positive constant coefficients one-dimensional operator.  To show solvability problem  use several important properties multinomial M...

We consider the standard hypergeometric differential operator $\mathcal D$ regarded as an on complex plane $\mathbb C$ and conjugate $\bar{\mathcal D}$. These operators formally commute are adjoint one to another with respect appropriate weight. find conditions when they in Nelson sense write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of Jac...