Discrete semiclassical orthogonal polynomials of class one
Diego Dominici, Francisco Marcellan
研究了s=1的离散半经典正交多项式。通过考虑Pearson方程的所有可能解，我们得到了五个典型族。我们还考虑这些和其他家庭的正交多项式之间的极限关系。
We study the discrete semiclassical orthogonal polynomials of class s=1. By considering all possible solutions of the Pearson equation, we obtain five canonical families. We also consider limit relations between these and other families of orthogonal polynomials.
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In this contribution, discrete semiclassical orthogonal polynomials of class $s\leq2$ are studied. By considering all possible solutions of the Pearson equation, we obtain the canonical families in each class. We also consider limit relations between these and other families of orthogonal polynomials.
A semiclassical perspective on multivariate orthogonal polynomials
研究了多元正交多项式的微分性质。本文考虑梯度满足某些拟正交条件的多元正交多项式。我们得到了这些多项式的几个特征，包括半经典Pearson微分方程的类似特征，结构关系和微分-差分方程.
The Symmetrical $H_{q}$-Semiclassical Orthogonal Polynomials of Class One
研究了对称$H_{q}$ -半经典正交$q$ -多项式的二次分解和对偶分类问题，其中$H_{q}$是Hahn算子.对于任何典型的情况下，递归系数，$q$ -模拟的皮尔逊型分布方程，矩和积分或离散表示。
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。
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。
The semiclassical--Sobolev orthogonal polynomials: a general approach
当多项式序列$(Q^{(λ)}_n)$关于内积$$ <p, r>_S=<{\bf u} ,{p\, r}> +λ<{\bf u}, {{\mathscr D}p \,{\mathscr D}r}>, $$正交时，我们说它是半经典Sobolev多项式序列，其中${\bf u}$是半经典线性泛函，${\mathscr D}$是微分，差分或$q$ -差分算子，$λ$是正常数。本文得到了这类多项式的代数和微分/差分性质以及它们与关于半经典泛函$\bf u$正交的多项式序列之间的代数关系。本文的主要目的是给出一个一般的方法来研究多项式正交于上述非标准内积，无论类型的运营商${\mathscr D}$考虑。最后，我们通过将这些结果应用于一些已知的Sobolev正交多项式族以及本文首次引入的一些新的Sobolev正交多项式族来说明我们的结果。
In this work we study orthogonal polynomials via polynomial mappings in the framework of the $H_q-$semiclassical class. We consider two monic orthogonal polynomial sequences $\{p_n (x)\}_{n\geq0}$ and $\{q_n(x)\}_{n\geq0}$ such that $$ p_{kn}(x)=q_n(x^k)\;,\quad n=0,1,2,\ldots\;, $$ being $k$ a fixed integer number such that $k\geq2$, and we prove that if one of the sequences $\{p_n (x)\}_{n\geq0}$ or $\{q_n(x)\}_{n\geq0}$ is $H_q-$semiclassical, then so is the other one. In particular, we show that if $\{p_n(x)\}_{n\geq0}$ is $H_q-$semiclassical of class $s\leq k-1$, then $\{q_n (x)\}_{n\geq0…
Epilegomena to the study of semiclassical orthogonal polynomials
本文针对Ismail关于Askey-Wilson算子结构关系刻画$\text{D}_q$-经典正交多项式的猜想，给出$\text{D}_q$-半经典正交多项式的两个刻画定理，重建标准导数与Askey-Wilson算子的对称性，构造二阶多项式序列证伪猜想并拓展至Hahn算子。
Second structure relation for $q$-semiclassical polynomials of the Hahn Tableau
q-哈恩表的q-经典正交多项式由它们的正交性条件和第一和第二结构关系来表征。不幸的是，对于q-半经典正交多项式（经典多项式的推广），我们只能在文献中找到第一个结构关系。本文推导了第二种结构关系。特别地，借助于q-半经典多项式列与其q-差列之间的一般有限型关系，得到了这样一种结构关系。
We consider Koornwinder's method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder's construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.
Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and study the corresponding second order difference equation. We suggest an Ansatz for its solutions playing the role of the semiclassical Green-Liouville Ansatz for solutions of the Schrödinger equation. The formulas obtained for $P_{n}(z)$ as $n\to\infty$ generalize the classical Bernstein-Szegö asymptotic formulas.
Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Lague…
Contiguous hypergeometric relations for semiclassical discrete orthogonal polynomials are described as Christoffel and Geronimus transformations. Using the Christoffel-Geronimus-Uvarov formulas quasi-determinatal expressions for the shifted semiclassical discrete orthogonal polynomials are obtained.
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $\pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of ge…
Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
本文导出了复平面上等值线上任意半经典测度的正交多项式所满足的微分方程组，以及通过改变这些测度所得到的变形方程的相容方程组。这些被证明保持相关的秩-2有理协变导数算子的广义单值性。相应的矩阵模型，由酉可对角化的矩阵与支持这些轮廓上的光谱进行了分析，它表明，所有相关的光谱曲线的系数是由对数导数的配分函数，或更一般地说，间隙概率。相关联的isomonodromic tau函数的重合，在一个明确计算的因素，与这些配分函数。
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。
An Introduction to the q-Laguerre-Hahn Orthogonal q-Polynomials
通过一个适当的q-差分方程，将q-Laguerre-Hahn形式的正交q-多项式作为q-半经典形式的推广来研究.给出类的概念和判定类的准则。得到了相应的形式Stieltjes级数所满足的q-Riccati方程。并建立了结构关系。突出了一些说明性的例子。
A basic class of symmetric orthogonal polynomials of a discrete variable
利用离散空间中Sturm-Liouville问题的一个推广，引入了一类带四个自由参数的离散变量对称正交多项式，它推广了所有经典的离散对称正交多项式.给出了这些多项式的标准性质，如二阶差分方程、多项式的显式、三项递推关系和正交关系。证明了从这类序列中可以抽取出两个具有20个不同权函数的超几何正交序列。此外，可以显式地计算与这些权函数相对应的矩。最后，详细研究了一个包含所有经典离散对称正交多项式的特例。
We present a method to obtain weight functions associated with linear and quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional of the orthogonal polynomial sequences in the Askey scheme, with the exception of the Jacobi, Bessel, Laguerre, and Hermite polynomials.
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…
Discrete orthogonal polynomials and difference equations of several variables
本文的目标是刻画所有以离散正交多项式为特征函数的多元二阶差分算子。在适当的假设下，我们给出了问题的完全解.
We introduce a new class of polynomials of multiple orthogonality with respect to the product of $r$ classical discrete weights on integer lattices with noninteger shifts. We give explicit representations in the form of the Rodrigues formulas. The case of two weights is described in detail.
In the paper we study the multipliers of Fourier series in polynomials orthogonal in continuous-discrete Sobolev's spaces. Multiplier Theorem for Fourier-Sobolev series is obtained. This result is based on the representation of the Fejér kernel, on the construction of the "humpbacked majorant" and weighted estimates of maximal functions.
Discrete orthogonal polynomials related to Hahn difference operator
本文研究线性格上的一般离散正交多项式，推导其结构关系及四阶差分方程，并分析半经典与经典特例，给出系数计算示例。
Differential equations for discrete Laguerre-Sobolev orthogonal polynomials
研究了质量点为零的离散Laguerre-Sobolev双线性型的正交多项式的微分性质.特别地，我们使用某些Casorati行列式构造正交多项式。利用这种构造，我们证明了它们是一个微分算子的本征函数（这将被明确地构造）。此外，该微分算子的阶是根据定义离散Laguerre-Sobolev双线性型的矩阵来明确计算的。
Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlevé equations
我们考虑单位圆上的正交多项式关于权是$q$ -Γ函数的商。我们表明，这些多项式的Verblunsky系数满足离散Painlevé方程，在一个Lax形式，这对应于一个$A_3^{(1)}$表面在Sakai的分类。
On discrete orthogonal polynomials of several variables
设$V$是$\RR^d$中的一组孤立点。在$V$上限制的实多项式空间上定义一个线性泛函$\CL$，$\CL f = \sum_{x \in V} f(x)ρ(x)$，其中$ρ$是$V$上的非零函数.多项式子空间，包含离散的正交多项式的双线性型$<f,g> = \CL(f g)$。一个结果表明，离散的正交多项式仍然满足三项关系和Favard定理在此一般设置。
Matrix-valued bispectral discrete orthogonal polynomials
本文构建了离散权重的矩阵值正交多项式统一构造，得到二阶差分算子的双谱序列，并给出Krawtchouk、Hahn、Meixner和Charlier多项式的矩阵模拟。
We obtain weight functions associated with $q$-linear and $q$-quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional for the Askey-Wilson polynomials and all the polynomial sequences in the q-Askey scheme, with the exception of the continuous $q$-Hermite polynomials.
The Kontorovich-Lebedev transform as a map between $d$-orthogonal polynomials
Kontorovich-Lebedev变换的一个微小修改是多项式向量空间上的自同构。这个$KL_α$ -变换在某些多项式序列上的作用将被讨论，并且将特别关注d-正交序列。例如，连续对偶哈恩多项式表现为拉盖尔型2-正交序列的$KL_α$ -变换。最后，对所有其$KL_α$ -变换是$d$ -正交序列的正交多项式序列进行了刻划：它们是满足特定条件的本质半经典多项式，并且$d$是偶数. Hermite和Laguerre多项式是这个问题的经典解。
We show how to get Krall polynomials from Krall discrete polynomials using a procedure of passing to the limit in some of the parameters of the family. We also show that this procedure has to be different to the standard one used in the Askey scheme to go from the classical discrete polynomials to the classical polynomials.
Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product
研究了一类包含离散Sobolev内积$\langle h,g \rangle = \int hg\, dμ+ \sum_{j=1}^m \sum_{i=0}^{N_j} M_{j,i} h^{(i)}(c_j) g^{(i)}(c_j)$的内积的正交多项式的渐近性质，其中$μ$是实直线上的某种复测度，$c_j$是$\supp(μ)$的补的复数.将Sobolev正交多项式与对应于测度$μ$的正交多项式进行比较。
We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyze relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the coefficients of the structure relations satisfied by these bivariate semiclassical orthogonal polynomials, also a matrix differential-difference equation for the bivariate orthogonal polynomials is deduced. The extension of the Painlevé equation for the coefficients of the three term relations to the bivariate case and a two dimensional version of the Langmui…
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 a companion paper [On semiclassical orthogonal polynomials via polynomial mappings, J. Math. Anal. Appl. (2017)] we proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations. In this work we use this fact to derive in an unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind. In particular we derive ordinary differential equations for these polynomials. As an application, we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of …
Boelen et al. (2010) deduced a $q$-discrete Painlevé equation satisfied by the recurrence coefficients of orthogonal polynomials and conjectured that the equation had a unique positive solution. We prove their conjecture and discuss related work in the area, highlighting a potentially unifying methodology.
We prove that all zeros of the polynomials orthogonal with respect to a measure $d μ(x;a) = d μ(x) + M δ(x-a)$, where $dμ$ is a nonatomic positive Borel measure and $M>0$, are increasing functions of the mass point $a$. Thus we solve partially an open problem posed by Mourad Ismail.
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.
Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials
我们给出四个例子的家庭的正交多项式的系数的递归关系满足离散Painlevé方程。第一个例子涉及弗洛伊德在实数线上的权重$|x|^ρ\exp(-|x|^m)$，我们重复弗洛伊德对$m=2,4,6$的推导和分析。当$m=4$对应于离散的Painlevé I方程时，Freud方程的递归系数。第二个例子处理单位圆上的正交多项式，用于权值$\exp(λ\cos θ)$。这些正交多项式在随机酉矩阵理论中有重要的应用。Periwal和Shevitz证明了递归系数满足离散Painlevé II方程。第三个例子是关于正整数的离散正交多项式。我们证明了广义Charlier多项式的递推系数可以从离散Painlevé II方程的解中得到。第四个例子处理$\pm q^n$上的正交多项式。本文考虑离散的$q$ -Hermite I多项式和某些离散的$q$ -Freud多项式，它们的递推系数满足离散Painlevé I的$q$ -变形.
Multiplicity of zeros and discrete orthogonal polynomials
我们考虑一个问题的边界的最大可能的多重性为零的一些扩展$\sum a_i F_i(x)$，在一定的点$c,$取决于所选择的家庭$\{F_i \}$。最重要的例子是一个多项式与$c=1.$它表明，这个问题自然会导致离散的正交多项式。使用这种连接，我们得到一些新的界限，特别是在一个多项式的零的多重性与规定的规范。
Special non uniform lattice ($snul$) orthogonal polynomials on discrete dense sets of points.
与多项式兼容的差分演算（即，使得应用于任何多项式的一阶均差算子必须产生较低次数的多项式）只能在当代$q-$演算中熟知的特殊格上进行。给出了在这种格上满足差分关系的正交多项式。特别是，被认为是密集的间隔（$|q|=1$）的晶格。
Extensions of discrete classical orthogonal polynomials beyond the orthogonality
众所周知，哈恩多项式族$\{h_n^{α,β}(x;N)\}_{n\ge 0}$关于直到$N$的某个权函数正交。本文给出了次数高于$N$的Hahn多项式的一种因式分解，并证明了这类多项式可由$Δ$ -Sobolev正交性刻画.我们还提出了一个类似的结果，双哈恩，Krawtchouk，和Racah多项式，并给出它们之间的极限关系的所有$n\in \XX N_0$。此外，为了得到Krawtchouk多项式的这个结果，我们将得到Meixner多项式的正交性的一个更一般的性质。
Some Appell-type orthogonal polynomials on lattices
本文研究q-二次格上Appell型正交多项式序列，通过刻画相应形式，给出Al-Salam Chihara多项式（含Rogers q-Hermite）的新特征，并为Askey-Wilson等算子推广到经典与半经典格上正交多项式提供方法。
Second order difference equations and discrete orthogonal polynomials of two variables
研究了二元二阶偏差分方程$ \CD u:= A_{1,1}(x) Δ_1 \nabla_1 u + A_{1,2}(x) Δ_1 \nabla_2 u + A_{2,1}(x) Δ_2 \nabla_1 u + A_{2,2}(x) Δ_2 \nabla_2 u & \qquad \qquad \qquad \qquad + B_1(x) Δ_1 u + B_2(x) Δ_2 u = λu, $，确定了它何时有正交多项式解。我们推导出$\CD$上的条件，使得权函数$W$存在，其中$W \CD u$是自伴的，并且差分方程具有关于$W$正交的多项式解。其解本质上是经典的二元离散正交多项式。
New sequences of discrete orthogonal polynomials associated with the modified Bessel function $K_μ(z)$ or Macdonald function are considered. The corresponding weight function is $λ^k ρ_{k+ν+1}(t)/ k!$, where $\ k \in \mathbb{N}_0, \ t \ge 0,\ ν> -1,\ 0 < λ< 1,\ ρ_μ(z) = 2 z^{μ/2} K_μ\left( 2\sqrt z\right)$. The limit case $t=0$ corresponds to the Meixner polynomials. Various properties, differential-difference recurrence relations are established. The modified sequence of polynomials with the weight $λ^k ρ_{k+ν+1}(λt)/ k! $ is investigated as well.
In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials $f_n^{(α)}(z)$ where $α$ is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and [{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}].
In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions.…
Preud's equations for orthogonal polynomials as discrete Painlevé equations
考虑关于指数权函数w（x）= exp（-P（x））的正交多项式p_n.许多人从Laguerre [49]开始研究递归系数的相关方程，以研究特殊的连分数、递归关系和各种渐近展开（G.弗洛伊德的贡献[28，56]）。最显著的例子是n = 2tw_n + w_n（w_n+1 + w_n + w_n-1），其中正交多项式的递归系数p_n+1 = xp_n-w_np_n-1与权w（x）= exp（-4（tx^3 + x^4））有关（[26，pp. 34-36]）。这个例子几乎出现在下面所有的参考文献中。这里描述了与离散Painlevé方程的联系。
The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, …
Given a finite set $F=\{f_1,\cdots ,f_k\}$ of nonnegative integers (written in increasing size) and a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant $\det(p_{f_i}(x+j-1))_{i,j=1,\cdots,k}$. In this paper we prove a nice invariant property for this kind of Casorati determinants when the set $F$ is changed by $I(F)=\{0,1,2,\cdots,\max F\}\setminus \{\max F-f:f\in F\}$. This symmetry is related to the existence of higher order difference equations for the orthogonal polynomials with respect to certain Chr…
On two families of discrete multiple orthogonal polynomials on a star-like set
本文研究星形集合上两类离散多重正交多项式（Charlier与Meixner权），探讨其升算子、Rodrigues公式、显式表达式及递推系数，并建立两族间的极限联系。
Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme
本文在$q^{-1}$-对称Askey框架下计算了五类正交多项式的双边离散与连续正交关系，并由$q^{-1}$-Askey--Wilson多项式的连续正交关系导出$q$-beta积分，其在$q\ o 1^{-}$极限下对应一个Ramanujan型beta积分并给出两种证明。
An electrostatic model for the roots of discrete classical orthogonal polynomials
本文提出一个静电模型，描述经典离散正交多项式根的行为，并将其推广至二阶线性差分方程多项式解的一般框架。
The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions -- the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical computations. In this paper we provide a partial solution to these problems. First, we construct an explicit basis of (non-orthogonal) eigenvectors of the discrete Fourier transform, thus extending the r…
Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
该研究探讨带纯虚位移的离散量子力学里多指标正交多项式的离散正交关系，验证了三类情形(1)多项式满足该关系并推测其归一化常数。
Toda and Laguerre-Freud equations for multiple discrete orthogonal polynomials with an arbitrary number of weights
本文将半经典多重离散正交多项式的研究扩展到任意数量权重的情形，并为多重广义Charlier族和多重广义Meixner II族推导出多重版本的Toda方程与Laguerre-Freud方程。
On a q-Deformation of the Discrete Painlevé I Equation and q-Orthogonal Polynomials
我提出了一个$q$ -模拟的离散Painlevé I方程，并在$q$ -正交多项式的特殊实现。
We consider multiple orthogonal polynomials with respect to Nikishin systems generated by two measures $(σ_1, σ_2)$ with unbounded supports ($\mbox{supp} \, σ_1 \subseteq \mathbb{R}_+$, $\mbox{supp} \, σ_2 \subseteq \,\mathbb{R}_-$) and $σ_2$ is discrete. A Nikishin type equilibrium problem in the presence of an external field acting on $\mathbb{R}_+$ and a constraint on $ \mathbb{R}_-$ is stated and solved. The solution is used for deriving the contracted zero distribution of the associated multiple orthogonal polynomials.
The discrete cosine transforms of types V--VIII are generalized to the antisymmetric and symmetric multivariate discrete cosine transforms. Four families of discretely and continuously orthogonal Chebyshev-like polynomials corresponding to the antisymmetric and symmetric generalizations of cosine functions are introduced. Each family forms an orthogonal basis of the space of all polynomials with respect to some weighted integral. Cubature formulas, which correspond to these families of polynomials and which stem from the developed discrete cosine transforms, are derived. Examples of three-dime…
Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles
这是“Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles：Announcement of Results”的完整版本，出现在此服务器上，也发表在IMRN 2003，第15号，第15页。821-858.
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