对称锥上的Laguerre函数与实情形下的递归关系

对称锥上Laguerre函数的微分递推关系

Differential Recursion Relations for Laguerre Functions on Symmetric Cones

设$Ω$是对称锥，$V$是相应的单欧氏若当代数.在\cite{ado,do,do04,doz2}中，我们考虑了$Ω$上的广义Laguerre函数族，它推广了$\mathbb{R}^+$上的经典Laguerre函数.这个族形成了$L^2(Ω,dμ_ν)$中的$L$不变函数的子空间的正交基，其中$dμ_ν$是锥上的某个测度，并且其中$L$是$V$上的线性变换的组，该线性变换使锥$Ω$不变并且将恒等式固定在$Ω$中。空间$L^2(Ω,dμ_ν)$支持作用在管域$T(Ω)=Ω+ iV.$上的全纯双同态群$G$的最高权表示。本文给出了$G$的李代数作用的一个显式公式，并通过这个作用确定了二阶微分算子，这些微分算子给出了广义Laguerre函数的微分递归关系，推广了经典的产生、保持、以及$\mathbb{R}^+$上Laguerre函数的湮灭关系。

Moment Representations of Type I X2 Exceptional Laguerre Polynomials

The $X_m$ exceptional orthogonal polynomials (XOP) form a complete set of eigenpolynomials to a differential equation. Despite being complete, the XOP set does not contain polynomials of every degree. Thereby, the XOP escape the Bochner classification theorem. In literature two ways to obtain XOP have been presented. When m=1, Gram-Schmidt orthogonalization of a so-called "flag" was used. For general m, the Darboux transform was applied. Here, we present a possible flag for the X_m exceptional Laguerre polynomials. We can write more about this. We only want to make specific picks when we also …

Moment Representations of the Exceptional $X_1$-Laguerre Orthogonal Polynomials

Exceptional orthogonal Laguerre polynomials can be viewed as an extension of the classical Laguerre polynomials per excluding polynomials of certain order(s) from being eigenfunctions for the corresponding exceptional differential operator. We are interested in the (so-called) Type I $X_1$-Laguerre polynomial sequence $\{L_n^α\}_{n=1}^\infty$, $\text{deg} \,p_n = n$ and $α>0$, where the constant polynomial is omitted. We derive two representations for the polynomials in terms of moments by using determinants. The first representation in terms of the canonical moments is rather cumbersome. We i…

多维海森堡卷积与多元Laguerre多项式的乘积公式

Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

设$p,q$为正整数。群$U_p(\b C)$和$U_p(\b C)\times U_q(\b C) $作为自同构群规范地作用在海森堡群$H_{p,q}:=M_{p,q}(\b C)\times \b R$上，其中$M_{p,q}(\b C)$是所有复$p\times q$ -矩阵的向量空间。相应的轨道空间可以分别用半正定矩阵的锥$Π_q$和Weyl腔$Ξ_q={x\in\b R^q: x_1\ge...\ge x_q\ge 0}$表示为$Π_q\times \b R$和$Ξ_q\times \b R$。本文计算了$Π_q\times \b R$和$Ξ_q\times \b R$上显式依赖于$p$的相关卷积。此外，我们扩展这些卷积的解析延拓到一系列的卷积结构的任意参数$p\ge 2q-1$。这导致$q\ge 2$在$Π_q\times \b R$上的非交换超群和在$Ξ_q\times \b R$上的交换超群的连续序列。在后一种情况下，我们描述的多元拉盖尔和贝塞尔函数$Π_q$和$Ξ_q$的对偶空间。特别地，我们给出了这些Laguerre函数在$Ξ_q$上的非正乘积公式。本文扩展了已知的情况$q=1$由于Koornwinder，Trimeche，和其他人，以及与整数$p$由于Faraut，Benson，Jenkins，Ratcliff，和其他人的情况下，组。此外，它与多元贝塞尔函数和…

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。

Complementary Polynomials From Rodrigues' Representations For Confluent And Hypergeometric Functions And More

Complementary polynomials of Legendre polynomials are briefly presented, as well as those for the confluent and hypergeometric functions, relativistic Hermite polynomials and corresponding new pre-Laguerre polynomials. The generating functions are all given in closed form and are much simpler than the standard ones. Some are simply polynomials in two variables. New recursions and addition formulas are derived.