Related papers: Integral representation for Jacobi polynomials and…
We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone $\mathbb{V}^{d+1}$ and its surface $\mathbb{V}^{d+1}_0$. To do so, we combine the theory of Jacobi polynomials on the cone…
In this note we give an explicit integral representation and an expanssion for the heat kernel $ H_n(t;x,y)$ associated to Fubini-Study Laplacians on quaternionic projective spaces $\mathbb{P}^n(\mathbb H)$, $n \geq 1$. This was possible by…
The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal…
We obtain sharp estimates for the Jacobi heat kernel in a range of parameters where the result has not been established before. This extends and completes an earlier result due to the authors. The proof is based on a generalization of the…
We study the even and odd Jacobi heat kernels defined in the context of the multidimensional double cone and its surface, the multidimensional hyperboloid and its surface, and the multidimensional paraboloid and its surface. By integrating…
We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat…
The rank two Jacobi algebra $\mathfrak{J}_2$ is identified as the dynamical algebra of the generic quadratic superintegrable model on the two-sphere. The physical representation of this algebra is obtained from its embedding in…
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This…
We prove heat kernel estimates for the $\bar\partial$-Neumann Laplacian acting in spaces of differential forms over noncompact, strongly pseudoconvex complex manifolds with a Lie group symmetry and compact quotient. We also relate our…
It is well-known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a…
A new algebraic approach for calculating the heat kernel for the Laplace operator on any Riemannian manifold with covariantly constant curvature is proposed. It is shown that the heat kernel operator can be obtained by an averaging over the…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
Infinitesimal holomorphic realizations for the Schr\"{o}dinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The…
In this paper, we study the heat equation associated with the Jacobi--Cherednik operator on the real line. We establish some basic properties of the Jacobi--Cherednik heat kernel and heat semigroup. We also provide a solution to the Cauchy…
This paper addresses a general method of polynomial transformation of hypergeometric equations. Examples of some classical special equations of mathematical physics are generated. Heun's equation and exceptional Jacobi polynomials are also…
The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an oscillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a…
In this survey article, we review the relation between heat kernels and path integrals. In particular, we review recent results on the approximation of the Wiener measure on compact manifold by measures on (finite-dimensional) spaces of…
We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric…
On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic one-forms, the Gaussian heat kernel upper…
We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series…