Roelof Koekoek
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.
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A generic differential operator on the vectorial space of polynomial functions was presented in a recent work and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Using the techniques therein developed, we prove an identity fulfilled by different differential operators and apply it in a systematic approach to the problem of finding polynomial eigenfunctions, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. In particular, we analyse a third order differential operator that does not increase the degr…
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 manuscript we study algebraic and analytic properties of the sequence of monic polynomials orthogonal with respect to a Jacobi differential operator. A fluid dynamics model for source points location of a flow of an incompressible fluid with preassigned stagnation points is also considered.
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic operator, we prove that a 2-orthogonal polynomial solution fulfils Hahn's property.
The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of spherical harmonics, algebraic and differential properties will be deduced.
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…
In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials. Several characterizations for these semi-classical functionals are given in terms of the corresponding (left) matrix orthogonal polynomials sequence. They involve a quasi-orthogonality property for their derivatives, a structure relation and a second order differo-differential equation. Finally we illustrate the preceding results with some non-trivial examples.
Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi--orthogonality conditions. We obtain several characterizations for these polynomials including the analogous of the semiclassical Pearson differential equation, the structure relation and a differential--difference equation.
In this paper, we derive new recurrence relations for two-variable orthogonal polynomials for example Jacobi polynomial, Bateman's polynomial and Legendre polynomial via two different differential operators $Ξ=\left(\frac{\partial }{\partial z} +\sqrt{w} \frac{\partial }{\partial w} \right)$ and $Δ=\left(\frac{1}{w} \frac{\partial }{\partial z} +\frac{1}{z} \frac{\partial }{\partial w} \right)$. We also derive some special cases of our main results.
Differential equations for a class of semiclassical orthogonal polynomials on the unit circle
本文研究单位圆上满足Pearson型微分方程的半经典正交多项式，通过结构关系和差分方程推导出一阶、二阶微分方程，并应用于推广的Jacobi多项式、修正Bessel多项式及离散Painlevé II方程。
Matrix valued orthogonal polynomials related to SU$(N+1)$, their algebras of differential operators and the corresponding curves
本文给出与SU$(N+1)$表示相关的矩阵值正交多项式族的两个微分算子代数例子，一个交换，一个非交换。
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.
On a moment generalization of some classical second-order differential equations generating classical orthogonal polynomials
本文构造了新的多项式系统作为泛函方程的解，推广了 Jacobi、Laguerre、Hermite 和 Bessel 经典正交多项式的二阶微分方程，并证明了多项式族与方程向经典情形的汇合收敛性。
Orthogonal polynomials of degree $n$ with respect to the weight function $W_μ(x) = (1-\|x\|^2)^μ$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ Δ- \la x, \nabla \ra^2 - (2 μ+d) \la x, \nabla \ra \right ] P = -n(n+2 μ+d) P $$ for $μ> -1$. The singular case of $μ= -1,-2, ...$ is studied in this paper. Explicit polynomial solutions are constructed and the equation for $ν= -2,-3,...$ is shown to have complete polynomial solutions if the dimension $d$ is odd. The orthogonality of the solution is also discussed.
Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect to a well-chosen parameter, according to principles established by D. G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in $a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$ of the orthogonal polynomials related to the weight $\exp(-x^4/4-tx^2)$ on {\blackb R\/} satisfy $4a_n^3\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$, and $a_n^2$ sa…
We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}ρ_{k}(z)\frac{d^k}{dz^k}, $$ where $\{ρ_k\}_{k=0}^{M}$ are complex polynomials such that $deg[ρ_k]\leq k, 0\leq k \leq M$, with equality for at least one index. We analyze the uniqueness and zero location of these polynomials. An interesting phenomenon occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator, we find a classification of the measures for which it …
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.
The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of orthogonal matrix polynomials whose elements are common eigenfunctions of several linearly independent second order differential operators. We show that the dual situation is also possible: there are examples of different families of matrix polynomials, each family orthogonal with respect to a different weight matrix, whose elements are eigenfunctions of a common …
Some examples of orthogonal matrix polynomials satisfying odd order differential equations
本文给出不可约化为标量权重的权重矩阵实例，其对应的矩阵正交多项式序列成为奇数阶线性微分算子的本征函数，突破了标量情形下阶数必须为偶数的限制。
In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.
Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for hyperexponential integrals. Further versions of this algorithm allow the computation of recurrence and differential equations from Rodrigues type formulas and from generating functions. In particular, these algorithms can be used to compute the differential/difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from th…
Orthogonal polynomials which are eigenfunctions of a partial differential operator
该研究确定了作为二阶或四阶线性微分算子本征函数的多变量包裹乘积正交多项式，补充了高维相关结果并给出三维情形的完整算子列表。
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。
We introduce the concept of $\D$-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+β_np_{…
Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory
本文提出一种将达布变换在算法上推广至线性微分系统张量积构造的方法，获得了三阶正交系统的显式变换及扩展至SL(2,C)-系统任意对称幂的框架，并在Maple中验证。
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。
In this paper, we obtain the ladder operators and associated compatibility conditions for the type I and the type II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann-Hilbert problems and the Christoffel-Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and…
We introduce a family of weight matrices $W$ of the form $T(t)T^*(t)$, $T(t)=e^{\mathscr{A}t}e^{\mathscr{D}t^2}$, where $\mathscr{A}$ is certain nilpotent matrix and $\mathscr{D}$ is a diagonal matrix with negative real entries. The weight matrices $W$ have arbitrary size $N\times N$ and depend on $N$ parameters. The orthogonal polynomials with respect to this family of weight matrices satisfy a second order differential equation with differential coefficients that are matrix polynomials $F_2$, $F_1$ and $F_0$ (independent of $n$) of degrees not bigger than 2, 1 and 0 respectively. For size $2…
Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family $\{p_ν(x)\}_{ν=0}^\infty$ orthogonal with respect to a measure supported on the real line …
In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types of identities for those functions. Some of these identities like differential equations, power series representations, and hypergeometric representations can even dealt with algorithmically, i.\ e.\ they can be computed by the Computer Algebra system, rather than only verified. The types of functions that can be treated by the given technique cover the genera…
We extend to a situation involving matrix valued orthogonal polynomials a scalar result that originates in work of Claude Shannon and a ground-breaking series of papers by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's. While these papers feature integral and differential operators acting on scalar valued functions, we are dealing here with integral and differential operators acting on matrix valued functions.
For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder's generalized Jacobi polynomials with four parameters $α,β\in\mathbb{N}_{0}$ and $M,N \ge 0$ determining the orthogonality measure on the interval $-1 \le x \le 1$. The corresponding differential equation of order $2α+2β+6$ is presented here as a linear combination of four elementary components which make the corresponding differential operator widely ac…
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville problem, are then introduced. First example generalizes the associated Legendre functions having extensive applications in physics and the second example introduces a generic differential equation with various sub-cases that have orthogonal solutions. For instance, this generic equation possesses a symmetric differential equation containing a basic solution of symm…
We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2\times 2$ introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel-Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of …
Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial sy…
Generalized semiclassical orthogonal polynomials on the unit circle: A Riemann-Hilbert perspective
该研究从黎曼-希尔伯特视角出发，推导单位圆上带权正交多项式及相关函数的微分方程，分析递推系数性质，并以广义修正Jacobi、Bessel权为案例验证方法。
We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=\int_{-1}^1f(x)g(x)(1-x^2)^αdx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big],$$ where $α>-1,$ $j\in \mathbb{N}\cup \{0\},$ and $M>0.$ Let $\{Q_n^{(α,M,j)}\}_{n\geq0}$ be the sequence of orthogonal polynomials with respect to the above inner product. These polynomials are eigenfunctions of a differential operator $\mathbf{T}. $ We establish the asymptotic behavior of the corresponding eigenvalues. Furthermore, we calculate the exact value $$r_0 = \lim_{n\rightarrow \infty}\frac{\log \lef…
Using the theory introduced by Casper and Yakimov, we investigate the structure of algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) on $\mathbb{R}$, and we derive algebraic and differential relations for these MVOPs. A particular case of importance is that of MVOPs with respect to a matrix weight of the form $W(x)=e^{-v(x)}e^{xA} e^{xA^\ast}$ on the real line, where $v$ is a scalar polynomial of even degree with positive leading coefficient and $A$ is a constant matrix.
Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable. More generally, normal forms are found. The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constr…
We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlevé equations and the differential equation is the $σ$-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlevé equations.
We introduce orthogonal polynomials $M_j^{μ,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $μ\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as $K$-finite vectors in the $L^2$-model of the minimal unitary representations of indefinite orthogonal groups, and reduce to the classical Laguerre polynomials $L_j^μ(x)$ for $\ell=0$. We establish various recurrence relations and integral representations for our polynomials, as well as a closed formula for the $L^2$-norm. Further we show that they are uniquely deter…
Orthogonal polynomials for the weight $x^ν \exp(-x - t/x),\ x, t > 0, ν\in \mathbb{R}$ are investigated. Differential-difference equations, recurrence relations, explicit representations, generating functions and Rodrigues-type formula are obtained.
We obtain a class of exact solutions of a Bessel-type differential equation, which is a six-parameter linear ordinary differential equation of the second order with irregular (essential) singularity at the origin. The solutions are obtained using the Tridiagonal Representation Approach (TRA) as bounded series of square integrable functions written in terms of the Bessel polynomial on the real line. The expansion coefficients of the series are orthogonal polynomials in the equation parameters space. We use our findings to obtain solutions of the Schrödinger equation for some novel potential fun…
A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we describe it in terms of the solution of a system of partial differential equations. In the case of two orthogonality measures the differential equation becomes ordinary. For Angelesco systems, the result is illustrated numerically.
We establish a symmetrization procedure in a context of general orthogonal expansions associated with a second order differential operator $L$, a `Laplacian'. Combined with a unified conjugacy scheme furnished in our earlier article it allows, via a suitable embedding, to associate a differential-difference `Laplacian' $\mathbb{L}$ with the initially given orthogonal system of eigenfunctions of $L$, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related `partial derivatives' decomposing $\mathbb{L}$ are skew-symmetric in an …
For a given polynomial $P$ with simple zeros, and a given semiclassical weight $w$, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of $P$. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of $P$. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of $P$ with respect to $w$ allows us to give more…
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…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete analogues. All the obtained solutions are expressed by way of special or elementary functions.
This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Jacobi polynomials which are orthogonal in a weighted Hilbert function space on the the interval (-1,+1) 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…
In this work, we investigate some connections between exact differential equations and harmonic functions and in particular, we obtain necessary and sufficient conditions for which exact equations admit harmonic solutions. As an application, we consider the orthogonal trajectories of harmonic functions, and among other results we obtain that the Cauchy-Riemann equations and the non-vanishing of the first partial derivatives are sufficient for any two curves to be orthogonal trajectories of each other. All curves throughout the work are restricted to the xy-plane.
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with two point masses at the endpoints of the interval of orthogonality. We show that such a differential equation is uniquely determined and we give explicit representations for the coefficients. In case of nonzero mass points the order of this differential equation is infinite, except for nonnegative integer values of (one of) the parameters. Otherwise, the fi…
We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses at the endpoints of the interval of orthogonality. In order to find explicit formulas for the coefficients of these differential equations we have to solve systems of equations involving derivatives of the classical Jacobi polynomials. We show that these systems of equations have a unique solution which is given explicitly. This is a consequence of the Jacobi…
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…
We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order equations was introduced by J. and R. Koekoek in 1999 essentially by using special function methods. In this paper, a completely elementary representation of the Jacobi-type differential operator of any even order is given. This enables us to trace the orthogonality relation of the Jacobi-type polynomials back to their differential equation. Moreover, we establ…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of isospectral matrices defined in terms of the zeros of such polynomials is discussed. The method involves reduction of certain partial differential, differential difference and differential q-difference equations to systems of ordinary differential equations and their subsequent linearization about the zeros of polynomials in question. Via this process, solvable …
The so-called exceptional orthogonal X1-polynomials arise as eigen functions of a Sturm-Liouville problem. In this paper, a generic classification of these polynomials is presented based on Pearson distributions family. Then, six special differential equations of the aforesaid classification are introduced and their polynomial solutions are studied in detail.
New orthogonal hybrid function based numerical method to solve system of fractional order differential equations
本文提出了一种基于新型正交混合函数的易实现且计算高效的数值方法，用于求解分数阶微分方程组。该方法利用广义一次性操作矩阵近似Riemann-Liouville分数阶积分，并通过实例验证了方法的有效性和可靠性。
A unified approach via Geronimus transformation to various types of orthogonal polynomials
该研究提出基于Geronimus变换的统一构造方法，以广义厄米特多项式为例推导各类正交多项式的零点位置、递推关系与微分方程等结果，覆盖多种正交多项式类型。
It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthermore, some well known results related to the Rodrigues operator are deduced. A more general characterization Theorem that the one given in [1] and [2] for the q-polynomials of the q-Askey and Hahn Table…
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