Differential Recursion Relations for Laguerre Functions on Symmetric Cones

Laguerre Functions on Symmetric Cones and recursion relations in the Real Case

In this article we derive differential recursion relations for the Laguerre functions on the cone C of positive definite real matrices. The highest weight representations of the group Sp(n,R) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain C+ i Sym(n,R). We then use the Laplace transform to carry the Lie algebra action over to L^2(C ,dm_t). The differential recursion relations result by restricting to a distinguished three dimensional subalgebra, which is isomorphic to sl(2,R).

Asymptotic computation of classical orthogonal polynomials

The classical orthogonal polynomials (Hermite, Laguerre and Jacobi) are involved in a vast number of applications in physics and engineering. When large degrees $n$ are needed, the use of recursion to compute the polynomials is not a good strategy for computation and a more efficient approach, such as the use of asymptotic expansions,is recommended. In this paper, we give an overview of the asymptotic expansions considered in [8] for computing Laguerre polynomials $L^{(α)}_n(x)$ for bounded values of the parameter $α$. Additionally, we show examples of the computational performance of an asymp…

Toda和Laguerre-Freud方程及超几何离散多重正交多项式的tau函数

Toda and Laguerre-Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials

本文研究离散多重正交多项式，通过tau函数和带状矩阵描述其性质，证明其提供Toda方程多分量可积扩展及Nijhoff-Capel方程的解，并探讨Laguerre-Freud方程在多种特例下的应用。

Monomial and Rodrigues orthogonal polynomials on the cone

We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{μ-\frac12}$, $μ> -\frac 12$, on the cone $\{(x,t): \|x\| \le t, \, x \in \mathbb{R}^d, t >0\}$ in $\mathbb{R}^{d+1}$. The first family consists of monomial polynomials $\mathsf{V}_{\mathbf{k},n}(x,t) = t^{n-|\mathbf{k}|} x^\mathbf{k} + \cdots$ for $\mathbf{k} \in \mathbb{N}_0^d$ with $|\mathbf{k}| \le n$, which has the least $L^2$ norm among all polynomials of the form $t^{n-|\mathbf{k}|} x^\mathbf{k} + \mathsf{P}$ with $°\mathsf{P} \le n-1$, and we will provide an explicit construction for …

广义 Charlier、广义 Meixner 与 Gauss 超几何正交多项式的 Laguerre-Freud 方程

Laguerre-Freud Equations for the Generalized Charlier, Generalized Meixner and Gauss Hypergeometric Orthogonal Polynomials

本文通过矩矩阵 Cholesky 分解研究三类离散正交多项式的 Laguerre-Freud 关系，并推导、比较相关非线性递推方程及其与 Painlevé 方程的联系。

Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Let $p,q$ positive integers. The groups $U_p(\b C)$ and $U_p(\b C)\times U_q(\b C) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\b C)\times \b R$ canonically as groups of automorphisms where $M_{p,q}(\b C)$ is the vector space of all complex $p\times q$-matrices. The associated orbit spaces may be identified with $Π_q\times \b R$ and $Ξ_q\times \b R$ respectively with the cone $Π_q$ of positive semidefinite matrices and the Weyl chamber $Ξ_q={x\in\b R^q: x_1\ge...\ge x_q\ge 0}$. In this paper we compute the associated convolutions on $Π_q\times \b R$ and $Ξ_q\times \b R$ explicitly dependin…