Related papers: Generalized Hermite Polynomials and the Heat Equat…

We consider a self-adjoint non-negative operator $H$ in a Hilbert space $\mathsf{L}^2(X,{\rm d}\mu)$. We assume that the semigroup $(\mathrm{e}^{-t H})_{t>0}$ is defined by an integral kernel, $p$, which allows an estimate of the form…

In this paper, we derive a Laplace-type integral representations for both the generalized Bessel function and the Dunkl kernel associated with the rank-two root system of type B_2. The derivation of the first one elaborates on the integral…

The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group of type B, contains the generic Hecke algebra for the symmetric group, i.e. the Weyl group of type A, as a subalgebra. Inducing the index representation of the…

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for…

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group $W$ of type $B_n$. We endow the polynomial ring ${\mathbb C} [x_1,\ldots\\\ldots, x_n]$ with a structure of…

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) L\'evy processes on half spaces for all $t>0$. These L\'evy processes may…

Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let…

Quantum differential operators on Reflection Equation Algebras, corresponding to Hecke symmetries R were introduced in previous publications. In the present paper we are mainly interested in quantum analogs of the Laplace and Casimir…

This paper deals with generalized Lipschitz spaces $\wedge^k_{\alpha,p,q}(\R)$ in the context of Dunkl harmonic analysis on $\R$, for all real $\alpha$. It also introduces a generalized Dunkl-Lipschitz spaces ${\cal…

Let $\Delta$ be the Laplace--Beltrami operator acting on a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$ with $m > n \ge 3$. Let $\frak{h}_t(x,y)$ be the kernels of the semigroup $e^{-t\Delta}$ generated by $\Delta$.…

We suggest a new technique of the asymptotic heat kernel expansion for minimal higher derivative operators of a generic $2M$-th order, $F(\nabla)=(-\Box)^M+\cdots$, in the background field formalism of gauge theories and quantum gravity.…

The main goal of this paper is to give potential theoretical approach to study the Dunkl Laplacian $\Delta_k$ which is a standard example of differential-difference operators. By introducing the Green kernel relative to $\Delta_k$, we prove…

In this article, we prove exact estimates for the $W$-invariant Dunkl kernel and heat kernel, for the root system of type $A$ with arbitrary positive multiplicities. We apply the estimates of the $W$-invariant Dunkl heat kernel to compute…

We establish the large-$N$ asymptotic expansion of the (central) trace of the heat kernel on any compact classical group $G_N\subset\mathrm{GL}_N(\mathbb{C})$, which extends a previous result known only for $\mathrm{U}(N)$ \cite{LM2}. It…

In this paper, we study linear transformations of the form $T[x^n]=P_n(x)$ where $\{P_n(x)\}$ is an orthogonal polynomial system. Of particular interest is understanding when these operators preserve real-rootedness in polynomials. It is…

In this paper, we present an explicit realization of q-deformed Calogero-Vasiliev algebra whose generators are first-order q-difference operators related to the generalized discrete q-Hermite II polynomials recently introduced in [13].…

We present an algebraic construction of the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement. These eigenfunctions are the superspace extension of the…

In this work, we introduce a new process by modifying the kernel in the time domain representation of the generalized Hermite process. This modification is constructed by means of multiplication of the kernel in the time definition of the…

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating…

We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle the heat-diffusion semigroup as a translation. With the given examples Kolmogorov-Riesz characterization of compact sets…