Diomedes Barcenas, Hugo Leiva, Yamilet Quintana, Wilfredo Urbina
In this paper we study the controllability of the controlled Laguerre equation and the controlled Jacobi equation. For each case, we found conditions which guarantee when such systems are approximately controllable on the interval $[0, t_1]$. Moreover, we show that these systems can never be exactly controllable
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In this paper using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials. We develop a transference method to obtain the $L^p$-continuity of the Gaussian-Riesz transform and the $L^p$-continuity of the Laguerre-Riesz transform from the $L^p$-continuity of the Jacobi-Riesz transform, in dimension one.
We show that for $δ$ below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesaro and Riesz means of order $δ$.
A transference result of the L^p continuity of the Jacobi Littlewood-Paley g-function to the Gaussian and Laguerre Littlewood-Paley g-function.
The $X_1$-Jacobi and the $X_1$-Laguerre exceptional orthogonal polynomials have been introduced and studied by Gómez-Ullate, Kamran and Milson in a series of papers. In this note, we establish some properties, such as interlacing, monotonicity with respect to the parameters and order, about the so-called \textit{regular} and \textit{exceptional} zeros of these two classes of polynomials.
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.
We will examine the electrostatic properties of exceptional and regular zeros of $X_m$-Laguerre and $X_m$-Jacobi polynomials. Since there is a close connection between the electrostatic properties of the zeros and the stability of interpolation on the system of zeros, we can deduce an Egerváry-Tur\' an type result as well. The limit of the energy on the regular zeros are also investigated.
We give the asymptotic distribution of the zeros of Jacobi-Piñeiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio asymptotics of multiple orthogonal polynomials. We show how these asymptotic zero distributions are related to the Fuss-Catalan distribution.
Orthogonality of the Jacobi and of Laguerre polynomials, P_n^(a,b) and L_n^(a), is established for a,b complex (a,b not negative integers and a+b different from -2,-3,...) using the Hadamard finite part of the integral which gives their orthogonality in the classical cases. Riemann-Hilbert problems that these polynomials satisfy are found. The results are formally similar to the ones in the classical case (when the real parts of a,b are greater than -1)
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials…
For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{(α_n, β_n)} (x), L_n^{(α_n)} (x),$\break $H_n^{(γ_n)} (x)$ the limit distributions of the zeros are found, when the sequences $α_n$ or $β_n$ tend to infinity with a larger order than $n$. The derivation uses special properties of the sequences in the corresponding recurrence formulae. The results are used to give second order approximations for the largest and smallest zero which improve (and generalize) the limit statements in a paper of Moak, Saff and Varga [11].
Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.
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。
Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra $\mathfrak{su}(1,1)$. The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of $\mathfrak{su}(1,1)$ are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and fun…
The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates f…
Orthogonality of quasi-nature spectral polynomials of Jacobi and Laguerre type
本文确定Jacobi多项式的一阶拟Christoffel和拟Geronimus多项式系数显式表达式，建立其拟谱多项式正交性，推导Laguerre型正交性，分析零点交错性质，并扩展至单位圆。
We derive an explicit expression for the kernel of the evolution group $\exp(-\mathrm{i} t H_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from $\ell^1$ to $\ell^\infty$ is given by $(1+t^2)^{-1/2}$.
In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be …
Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
本文针对离散量子力学等谱达布变换的II型构造方法，提出了II型多指标小$q$-雅可比与小$q$-拉盖尔正交多项式，补充了该领域基于II型构造的正交多项式研究成果。
An Askey Scheme for Jacobi Systems of Mixed Type: Laguerre Limits of the First and Second Kinds and Hermite Limits
本文为混合型Jacobi多重正交系统建立Askey汇合方案，通过变量重标度导出两类Laguerre及Hermite极限系统，给出显式极限权重、正交形式与矩收敛，并证明三条路径得到一致结果。
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree $\ell=1,2,...$, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change.
Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-Pöschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner …
We provide analytic proofs for the shape invariance of the recently discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux-Pöschl-Teller potential by a degree \ell (\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3\ell involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by…
We study the monotonic behaviour of the zeros of the multiple Jacobi-Angelesco orthogonal polynomials, in the diagonal case, with respect to the parameters $α,β$ and $γ$. We prove that the zeros are monotonic functions of $α$ and $γ$ and consider some special cases of how the zeros depend on $β$, especially in the presence of symmetry. As a consequence we obtain results about monotonicity of zeros of Jacobi-Laguerre and Laguerre-Hermite multiple orthogonal polynomials too.
As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials in the framework of `discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual stat…
We present a new set of infinitely many shape invariant potentials and the corresponding exceptional (X_{\ell}) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one dimensional quantum mechanics and the corresponding X_{\ell} Laguerre and Jacobi polynomials (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417). The new X_{\ell} Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X_{\ell} Jacobi polynomials and the potentials, whereas the known X_{\ell} L…
The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A generalizatio…
The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition
该研究证明了改进的Koornwinder-Kostenko-Teschl猜想，通过结合中心围道估计等方法得到雅可比函数的界，将其应用于离散拉盖尔演化得到最优正参数衰减，确定了长时间衰减指数的最优值。
In this article, the study of the orthogonality properties of $q$-polynomials of the Hahn class started in the initial article by R. Álvarez-Nodarse, R. Sevinik-Adıgüzel, and H. Taşeli, \textit{On the orthogonality of $q$-classical polynomials of the Hahn class I} is proceeded. To be more specific, the orthogonality properties of the $q$-polynomials belonging to the $\emptyset$-Hermite-Laguerre/Jacobi, $\emptyset$-Jacobi/Hermite-Laguerre, 0-Laguerre/Jacobi-Bessel and 0-Jacobi/Laguerre-Bessel cases are studied by taking into account the idea considered in the initial paper. In particular, a new…
We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 684} (2010), 173-176]. These $X_{\ell}$ polynomials are global solutions of second order Fuchsian differential equations with $\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape in…
On the extremal eigenvalues of Jacobi ensembles at zero temperature
本文研究了零温度下Jacobi族极值本征值的极限，通过Bessel函数推导了冻结情况下的协方差极限，对应于硬边分析和有限β的结果。
Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials
本文证明多指标 Laguerre 与 Jacobi 多项式虽破坏三项递推，仍满足离散正交关系，并识别低次补多项式、确定 Christoffel 数。
We study Jacobi operators $J_{p}$, $p> -1$, whose eigenfunctions are Laguerre polynomials. All operators $J_{p}$ have absolutely continuous simple spectra coinciding with the positive half-axis. This fact, however, by no means imply that the wave operators for the pairs $J_{p}$, $J_{q}$ where $p\neq q$ exist. Our goal is to show that, nevertheless, this is true and to find explicit expressions for these wave operators. We also study the time evolution of $(e^{-J t} f)_{n}$ as $|t|\to\infty$ for Jacobi operators $J$ whose eigenfunctions are different classical polynomials. For Laguerre polynomi…
Ladder Operators for Laguerre-type and Jacobi-type Orthogonal Polynomials
本文通过黎曼-希尔伯特问题和两种推导策略，推导了雅可比型正交多项式在参数>-1时的拉格朗日算子，并验证了结论的正确性。
Connecting Exceptional Orthogonal Polynomials of Different Kind
本文将经典 Jacobi、Laguerre、Hermite 正交多项式间的渐近极限与二次变换推广到同余维的例外正交多项式，建立不同种类 $X_m$ 多项式间的系统联系。
In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which give a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez-Ullate et al. to derive explicit expressions for the coefficients of the recurrence relation…
We find all polynomials solutions $P_n(x)$ of the abstract "hypergeometric" equation $L P_n(x) = λ_n P_n(x)$, where $L$ is a linear operator sending any polynomial of degree $n$ to a polynomial of the same degree with the property that $L$ is two-diagonal in the monomial basis, i.e. $L x^n = λ_n x^n + μ_n x^{n-1}$ with arbitrary nonzero coefficients $λ_n, μ_n$ . Under obvious nondegenerate conditions, the polynomial eigensolutions $L P_n(x) = λ_n P_n(x)$ are unique. The main result of the paper is a classification of all {\it orthogonal} polynomials $P_n(x)$ of such type, i.e. $P_n(x)$ are ass…
Coefficients in the expansions of the form $\partial P_{n}(λ;z)}/\partialλ=\sum_{k=0}^{n}a_{nk}(λ)P_{k}(λ;z)$, where $P_{n}(λ;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi) orthogonal polynomial of variable $z$ and $λ$ is a parameter, are evaluated. A method we adopt in the present paper differs from that used by Fröhlich [Integral Transforms Spec. Funct. 2 (1994) 253] for the Jacobi polynomials and by Koepf [Integral Transforms Spec. Funct. 5 (1997) 69] for the generalized Laguerre and the Gegenbauer polynomials.
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.
The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…
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…
We present determinantal formulas for families of exceptional $X_m$-Laguerre and exceptional $X_m$-Jacobi polynomials and also for exceptional $X_2$-Hermite polynomials. The formulas resemble Vandermonde determinants and use the zeros of the classical orthogonal polynomials.
Muttalib--Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form $\prod_{1 \le j < k \le N}(λ_k - λ_j) (λ_k^θ- λ_j^θ)$. We study the Laguerre and Jacobi versions of this model --- so named by the form of the one-body interaction terms --- and show that for $θ\in \mathbb Z^+$ they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general $θ> 0$, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent …
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.
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree $\ell=1,2,...$, and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new $X_\ell$ polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which th…
Wigner- and Marchenko-Pastur-type limits for Jacobi processes
本文研究 BC 型根系驱动的 Jacobi 过程，利用随机微分方程推导出当粒子数趋于无穷时经验分布的 Wigner 半圆律和 Marchenko-Pastur 律的几乎必然极限，并推广了 Jacobi 系综及多项式零点的相关平稳极限结果。
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.
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。
The exceptional Racah and q-Racah polynomials are constructed. Together with the exceptional Laguerre, Jacobi, Wilson and Askey-Wilson polynomials discovered by the present authors in 2009, they exhaust the generic exceptional orthogonal polynomials of a single variable.
Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 <A, x_k >B,$ which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with the relative precision, roughly speaking, $O(k^{-2/3}).$
An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality
该研究将 Wolfs 的 Jacobi 型和 Laguerre 型多重正交系统嵌入统一的 Askey 型合流方案，通过 Bernstein 变换构造 Hahn 型祖先，并导出多种正交多项式族及其合流关系。
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