Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on \b R^N. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group $S_N$, our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.Comment: 24 pages, AMS-LaTe

A positive radial product formula for the Dunkl kernel

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for non-negative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial hereby means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.Comment: 25 page

Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type $BC$. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds $G/K$ over one of the (skew) fields $\mathbb F= \mathbb R, \mathbb C, \mathbb H.$ We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of $K$-biinvariant functions on $G$

Olshanski spherical functions for infinite dimensional motion groups of fixed rank

Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$ associated with motion groups over the fields $\mathbb F=\mathbb R,\mathbb C,\mathbb H$ with $p\geq q$ and fixed $q$ as well as the inductive limit $p\to\infty$,the Olshanski spherical pair $(G_\infty,K_\infty)$. We classify all Olshanski spherical functions of $(G_\infty,K_\infty)$ as functions on the cone $\Pi_q$ of positive semidefinite $q\times q$-matrices and show that they appear as (locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\to\infty$. The latter are given by Bessel functions on $\Pi_q$. Moreover, we determine all positive definite Olshanski spherical functions and discuss related positive integral representations for matrix Bessel functions. We also extend the results to the pairs $(M_{p,q} \rtimes (U_p\times U_q),(U_p\times U_q))$ which are related to the Cartan motion groups of non-compact Grassmannians. Here Dunkl-Bessel functions of type B (for finite $p$) and of type A (for $p\to\infty$) appear as spherical functions

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a sharp Mehler-Heine formula, that is an approximation of the Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of $G/K$, which are constructed by successive decompositions of tensor powers of spherical representations of $G$. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases