J. Koekoek, R. Koekoek, H. Bavinck
We obtain all spectral type differential equations satisfied by the Sobolev-type Laguerre polynomials. This generalizes the results found in 1990 by the first and second author in the case of the generalized Laguerre polynomials defined by T.H. Koornwinder in 1984.
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C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted $L^{p}$ spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre--Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B. Xh. Fejzullahu and F. Marcellán.
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrix which defines the discrete Laguerre-Sobolev bilinear form.
In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of the classical Laguerre polynomials and deduce the coefficients of their corresponding five-term recurrence relation, as well as the asymptotic behavior of these coefficients when the degree of the polynomials tends to infinity. Also, the outer relative asymptotics of or…
The Sobolev-Laguerre polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses $M,N > 0$ at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter $α\in\mathbb{N}_{0}$, a spectral differential equation of finite order $4α+10$. In this paper we establish a new explicit representation of the corresponding differential operator which cons…
In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev ty…
Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain
本文提出了一种基于Laguerre-Sobolev正交多项式的谱方法，用于求解具有奇异势的Dirichlet问题，实现了高效递归计算并验证了谱精度。
Electrostatic models for zeros of Laguerre-Sobolev polynomials
本文研究 Laguerre-Sobolev 正交多项式，建立其与标准 Laguerre 多项式的联系，推导阶梯算子和二阶微分方程，并给出零点静电模型成立的充分条件及实例。
The solution of conformable Laguerre differential equation using conformable Laplace transform
本文利用合形拉普拉斯变换求解合形拉盖尔及关联拉盖尔微分方程，验证解与幂级数解一致，还讨论了拉盖尔多项式性质并提出证明合形Rodriguez公式与生成函数。
In the enduring, fruitful research on spectral differential equations with polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are playing a prominent role. Being orthogonal on the positive half-line with respect to the Laguerre weight and an additional point mass $N \ge 0$ at the origin, these polynomials satisfy, for any $α\in\mathbb{N}_{0}$, a linear differential equation of order $2α+4$. In the present paper we establish a new elementary representation of the corresponding 'Laguerre-type' differential operator and show its symmetry with respect to the underlying weigh…
In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the context of Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials, are notoriously difficult to compute. In this paper, we propose the use of an alternative expansion, due to Buchholz, in terms of Bessel functions of the first kind. The coefficients in this expansion can be obtained in a straightforward way using symbolic computation. As an application, w…
Let $Ω$ be a symmetric cone and $V$ the corresponding simple Euclidean Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of generalized Laguerre functions on $Ω$ that generalize the classical Laguerre functions on $\mathbb{R}^+$. This family forms an orthogonal basis for the subspace of $L$-invariant functions in $L^2(Ω,dμ_ν)$, where $dμ_ν$ is a certain measure on the cone and where $L$ is the group of linear transformations on $V$ that leave the cone $Ω$ invariant and fix the identity in $Ω$. The space $L^2(Ω,dμ_ν)$ supports a highest weight representation of the group $G$ o…
On higher-order differential equations for Laguerre-Hahn orthogonal polynomials
本文提出一种构造性算法，基于结构关系层级体系，为Laguerre-Hahn正交多项式推导任意阶齐次线性微分方程，并应用于两类零类族，给出五阶和十阶显式方程。
Orthogonality of a new family of $q$-Sobolev type polynomials
本文构造了源于 little $q$-Jacobi 与 $q$-Laguerre 多项式的新 $q$-多项式，证明其具有积分表示、三阶 $q$-差分方程和四项递推关系，并定义一般 $q$-Sobolev 型正交性，分析了特例、参数极限、零点及交错性。
Sequentially-ordered Sobolev inner product and Laguerre-Sobolev polynomials
本文研究一般离散 Sobolev 型内积下的正交多项式，证明其零点在支撑凸包内部至少有 n-d* 个，并在经典 Laguerre 测度情形下给出外部相对渐近。
Let $μ$ be a finite positive Borel measure supported on R, $\LL[f] =xf''+(α+1-x)f'$ with $α>-1$, or $\LL[f] =\frac{1}{2}f''-xf'$, and $m$ a natural number. We study algebraic, analytic and asymptotic properties of the sequence of monic polynomials $\{Q_n\}_{n>m}$ orthogonal with respect to the operator $\LL$. We also provide a fluid dynamics model for the zeros of these polynomials.
On a general method for deriving a fourth-order differential equation satisfied by Laguerre-Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite
本文提出一种构造性方法，通过结构关系、逐次导数和代数消元技术，推导Laguerre-Hahn正交多项式序列满足的四阶线性微分方程，并给出类0的新结果。
$2\times 2$ Laguerre-type differential operator with triangular eigenvalue
本文完整刻画了在 $(0,\infty)$ 上关于 $2\times 2$ 不可约权对称且具有下三角特征值的所有 Laguerre 型微分算子，并给出了三类参数化显式族及一个单参数新表达式。
We study algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) with respect to a weight matrix of the form $W^{(ν)}_ϕ(x) = x^νe^{-ϕ(x)} W^{(ν)}_{pol}(x)$, where $ν>0$, $W^{(ν)}_{pol}(x)$ is certain matrix valued polynomial and $ϕ$ an entire function. We introduce a pair differential operators $\mathcal{D}$, $\mathcal{D}^{\dagger}$ which are mutually adjoint with respect to the matrix inner product induced by $W^{(ν)}_ϕ(x)$. We prove that the Lie algebra generated by $\mathcal{D}$ and $\mathcal{D}^{\dagger}$ is finite dimensional if and only if…
We prove a necessary and sufficient condition for the integrability of the weight associated to the exceptional Laguerre polynomials. This condition is very much related to the fact that the associated second order differential operator has no singularities in $(0,+\infty)$.
In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: $L y + λ_n D y = 0$, where $L,D$ are linear differential operators with polynomial coefficients not depending on $n$. For positive integer values of the parameters $r,c$ these polynomials are Sobolev orthogonal polyn…
This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Laguerre polynomials which are orthogonal in a weighted Hilbert function space on the positive half-line of the real line. These polynomials are generated by a second-order ordinary linear differential equation with a spectral parameter. Some additional information on the Sturm-Liouville form of this equation is given in this note, together with details of the singular differential operators generated in the weighted Hilbert function space. In particular, structured boundary conditions are given to determine the spe…
The linearization problem for nonlinear second-order ODEs to the Laguerre form by means of generalized Sundman transformations (S-transformations) is considered, which has been investigated by Duarte et al. earlier. A characterization of these S-linearizable equations in terms of first integral and procedure for construction of linearizing S-transformations has been given recently by Muriel and Romero. Here we give a new characterization of S-linearizable equations in terms of the coefficients of ODE and one auxiliary function. This new criterion is used to obtain the general solutions for the…
We study the bispectrality of Laguerre type polynomials, which are defined by taking suitable linear combinations of a fixed number of consecutive Laguerre polynomials. These Laguerre type polynomials are eigenfunctions of higher-order differential operators and include, as particular cases, the Krall-Laguerre polynomials. As the main results, we prove that these Laguerre type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). We also prove that the Krall-Laguerre families are the only polynomials which are orthogonal with respect to a measure on the real…
We consider the discrete Sobolev inner product $$(f,g)_S=\int f(x)g(x)dμ+Mf^{(j)}(c)g^{(j)}(c), \quad j\in \mathbb{N}\cup\{0\}, \quad c\in\mathbb{R}, \quad M>0, $$ where $μ$ is a classical continuous measure with support on the real line (Jacobi, Laguerre or Hermite). The orthonormal polynomials with respect to this Sobolev inner product are eigenfunctions of a differential operator and obtaining the asymptotic behavior of the corresponding eigenvalues is the principal goal of this paper.
The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. We extend the Laguerre calculus for nilpotent groups of step two, and test it in the determining of the fundamental solution of the sub-Laplace operator. We also apply it to find the Szegö kernels of the projection operators to a kind of regular functions on the quaternion Heisenberg group.
Applications of Laguerre transform to solve Schrödinger-type equations and Differential Equations of order four
本文提出应用有限Laguerre变换，结合基础线性代数方法，求解高于二阶的微分方程及一维稳态Schrödinger方程。
In 1999, Grunbaum, Haine and Horozov defined a large family of commutative algebras of ordinary differential operators which have orthogonal polynomials as eigenfunctions. These polynomials are mutually orthogonal with respect to a Laguerre-type weight distribution, thus providing solutions to Krall's problem. In the present paper we give a new proof of their result which establishes a conjecture, concerning the explicit characterization of the dual commutative algebra of eigenvalues. In particular, for the Koornwinder's generalization of Laguerre polynomials, our approach yields an explicit s…
We present results on co-recursive associated Laguerre and Jacobi polynomials which are of interest for the solution of the Chapman-Kolmogorov equations of some birth and death processes with or without absorption. Explicit forms, generating functions, and absolutely continuous part of the spectral measures are given. We derive fourth-order differential equations satisfied by the polynomials with a special attention to some simple limiting cases.
Notes on $q$-partial differential equations for $q$-Laguerre polynomials and little $q$-Jacobi polynomials
该研究定义两类齐次$q$-正交多项式，证明其与$q$-偏微分方程组解的等价性，推广了两类经典$q$-积分，并给出便于恒等式计算的$q$-拉盖尔多项式算子表示。
Matrix exceptional Laguerre polynomials
本文利用二阶微分算子分解与 Darboux 变换构造矩阵值例外多项式，分析其正交性与稠密性，并给出任意大小的矩阵值例外 Laguerre 多项式示例。
This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre secondorder differential equations, and on the fourth-order Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differ…
On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities
本文研究了仅有实零点的实多项式 $p$ 的微分多项式 $F_{\varkappa}[p](z)$ 的非实零点个数上下界，构造反例推翻了 B. Shapiro 关于 $F_{\tfrac{n-1}{n}}[p](z)$ 实零点个数的猜想，并提出了推广 Hawaii 猜想的新猜想。
Sobolev orthogonal polynomials: Connection formulae
本文研究了关于离散 Sobolev 内积正交的多项式序列，推导出若干连接公式，并具体探讨了 Laguerre 情形下的应用。
The spectral order on $\bR^n$ induces a natural partial ordering on the manifold $\calH_{n}$ of monic hyperbolic polynomials of degree $n$. We show that all differential operators of Laguerre-Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space $\li$ of real bounded sequences. As a consequence, we deduce that the monoid $\calA'$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre-Pólya type which are both ex…
In this paper we deal with a family of non--standard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so--called $Δ$--Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the $Δ$--Meixner--Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for L…
Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$ \langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y) \,dx\, dy + λf(c_1,c_2)g(c_1,c_2) $$ are constructed, where $w_i$ is a weight function on $[a_i,b_i]$ for $i = 1, 2$, $λ> 0$, and $(c_1, c_2)$ is a fixed point. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions, which serve as primary e…
The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painleve transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painleve equation. The methodology used to derive these results rely on two theories regarding differential equations for orthogonal polynomial systems, one involving isomonodromic deformations and the other ladder operators. We compare the two theories…
We look for spectral type differential equations for the generalized Jacobi polynomials found by T.H. Koornwinder in 1984 and for the Sobolev-Laguerre polynomials. We introduce a method which makes use of computeralgebra packages like Maple and Mathematica and we will give some preliminary results.
In this paper we study modified kernel polynomials: $u_n(x) = \sum_{k=0}^n c_k g_k(x)$, depending on parameters $c_k>0$, where $\{ g_k \}_0^\infty$ are orthonormal polynomials on the real line. Besides kernel polynomials with $c_k = g_k(t_0)>0$, for example, $c_k$ may be chosen to be some other solutions of the corresponding second-order difference equation of $g_k$. It is shown that all these polynomials satisfy a $4$-th order recurrence relation. The cases with $g_k$ being Jacobi or Laguerre polynomials are of a special interest. Suitable choices of parameters $c_k$ imply $u_n$ to be Sobolev…
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that when the Jacobi parameters $α$ and $β$ are nonnegative integers the Jacobi-Sobolev orthogonal polynomials are eigenfunctions of a differential operator of finite order (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of…
Differential and Difference Equations for Recurrence Coefficients of Orthogonal Polynomials with a Singularly Perturbed Laguerre-type Weight
本文针对奇异扰动 Laguerre 型权函数下的首一正交多项式，采用阶梯算子方法建立递推系数的非线性二阶差分方程及微分-差分方程，并据此得到其大 n 渐近展开。
On Differential Systems in Sobolev spaces with Generic Inhomogeneous Boundary Conditions
本文在Sobolev空间中研究了具一般非齐次边界条件的任意阶常微分方程线性系统，证明了边值问题的Fredholm性质并求出核与余核维数，同时给出解关于参数连续的充要条件。
In this paper we study the asymptotics (as $n\to \infty$) of the sequences of Laguerre polynomials with varying complex parameters $α$ depending on the degree $n$. More precisely, we assume that $α_n = n A_n, $ and $ \lim_n A_n=A \in \mathbb{C}$. This study has been carried out previously only for $α_n\in \mathbb{R}$, but complex values of $A$ introduce an asymmetry that makes the problem more difficult. The main ingredient of the asymptotic analysis is the right choice of the contour of orthogonality, which requires the analysis of the global structure of trajectories of an associated quadrat…
Matrix valued Laguerre polynomials are introduced via a matrix weight function involving several degrees of freedom using the matrix nature. Under suitable conditions on the parameters the matrix weight function satisfies matrix Pearson equations, which allow to introduce shift operators for these polynomials. The shift operators lead to explicit expressions for the structures of these matrix valued Laguerre polynomials, such as a Rodrigues formula, the coefficients in the three-term recurrence, differential operators, and expansion formulas.
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).
Boundedness of bilinear pseudo-differential operators with $BS^m_{0,0}$ symbols on Sobolev spaces
本文研究双线性Hörmander类$BS^m_{0,0}$象征对应的伪微分算子，借助象征演算和$S_{0,0}$型算子的Lebesgue空间有界性，证明其在Sobolev空间上的有界性。
We investigate the type I and type II multiple orthogonal polynomials on an $r$-star with weight function $|x|^βe^{-x^r}$, with $β>-1$. Each measure $μ_j$, for $1\leq j \leq r$, is supported on the semi-infinite interval $[0,ω^{j-1}\infty)$ with $ω=e^{2πi/r}$. For both the type I and the type II polynomials we give explicit expressions, the coefficients in the recurrence relation, the differential equation and we obtain the asymptotic zero distribution of the polynomials on the diagonal. Also, we give the connection between the Laguerre-Angelesco polynomials and the Jacobi-Angelesco polynomial…
We study negative powers of Laguerre differential operators in $\R$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates, with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution ty…
We look for spectral type differential equations for the generalized Jacobi polynomials and for the Sobolev-Laguerre polynomials. We use a method involving computeralgebra packages like Maple and Mathematica and we will give some preliminary results.
We prove local and global invertibility of Sobolev solutions of certain differential inclusions which prevent the differential matrix from having negative eigenvalues. Our results are new even for quasiregular mappings in two dimensions.
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type, Legendre-type.
Limit theorems for differential systems with multipoint boundary conditions in Sobolev spaces
研究Sobolev空间中带多点边界条件且含Caputo导数的线性常微分方程组边值问题，建立解关于参数在该空间中连续的构造性充分条件。
A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.
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 …
A description of the Laguerre-Hahn orthogonal polynomials of class zero revisited
本文完整描述了零类Laguerre-Hahn形式，包括两个类似Bessel的新族，并修正了所有族的结构关系，同时通过符号计算推导出其中一个新族的四个新结构关系和一个四阶微分方程。
We consider the Fredholm one-dimensional boundary-value problems in Sobolev spaces.We have obtained several important results about the indixes of functional operators, the criterion of their correct well-posedness, the criterion of the continuous dependence of the solutions of these problems on the parameter, the degree of convergence of these solutions, and sufficient constructive conditions under which the solutions of the most general class of multipoint boundary-value problems are continuous with respect to the parameter. Eeach of these boundary-value problems corresponds to a certain rec…
Hardy spaces and Campanato spaces associated with Laguerre expansions and higher order Riesz transforms
本文针对拉盖尔微分算子\mathcal{L}_ν，建立了全范围\\\\(p \in (0,1]\\\\)的相关哈代空间理论，证明了对应BMO型空间是其对偶空间，并验证了高阶里斯变换在勒贝格空间等上的有界性。
We shall give bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class and satisfying a second order differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polinomials, which are uniform in all the parameters.
Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain parameters of type that are usually restricted to a half-line, or a product of half-lines if higher dimensions are considered. Following earlier research done by Hajmirzaahmad, we deal in this paper with Laplacians in the above-mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the cla…
The linearization problem of a second-order ordinary differential equation by the generalized Sundman transformation was considered earlier by Duarte, Moreira and Santos using the Laguerre form. The results obtained in the present paper demonstrate that their solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation is not complete. We also give examples which show that the Laguerre form is not sufficient for the linearization problem via the generalized Sundman transformation.
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