Rodica Simion, Dennis W. Stanton
Three specializations of a set of orthogonal polynomials with ``8 different q's'' are given. The polynomials are identified as $q$-analogues of Laguerre polynomials, and the combinatorial interpretation of the moments give infinitely many new Mahonian statistics on permutations.
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The 3-term recurrence relation for Hermite polynomials was recently generalized to a recurrence relation for Wronskians of Hermite polynomials. In this note, a similar generalization for Laguerre polynomials is obtained.
In this paper, the structures to a family of biorthogonal polynomials that approximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the generalized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from t…
I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x \le 0$.
By utilizing a two-dimensional extension of a very general series transform given by Bailey, Exton [Indian J. pure appl. Math. 24 (6) (1993), 401-408] deduced a very general double generating relation of a product of a pair of Laguerre polynomials and obtained a number of useful relations with elementary functions, Bessel functions, Hermite polynomials and single series expansions of pairs of Laguerre polynomials. Unfortunately, some of the results given by Exon contain errors and thus this is the aim of this short note to provide the corrected form of these results.
We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells.
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 introduce new families of q-deformed 2D Laguerre-Gould-Hopper polynomials. For these polynomials we establish connection formulae which extend some known ones.
In this paper we introduce a version of irreducible Laguerre polynomials in two variables and prove for it a congruence property, which is similar to the one obtained by Carlitz for the classical Laguerre polynomials in one variable.
The bilinear generating function for products of two Laguerre 2D polynomials Lm;n(z; z0) with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper…
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.
Motivated by the work of Prajapati \emph{et al.} \cite{PAA}, here we study some explicit form of the generalized Laguerre polynomials $L_{\lfloor\frac{n}{q}\rfloor}^{(α,β)}(z)$, when $q=1$.
We extend the notion of general translation operator to exceptional Laguerre polynomials. To this we investigate the associated singular hyperbolic Cauchy problem. We derive a maximum principle with respect to this Cauchy problem and applying it we determine the norm of the translation operator. As an application we give Nikol'skii inequalities with respect to exceptional Laguerre polynomials.
Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how certain sums of such polynomials with a fixed total degree relate to some standard polynomials.
We derive a system of difference equations satisfied by the three-term recurrence coefficients of some families of discrete orthogonal polynomials.
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…
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 $(0,+\infty)$.
The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way, and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials we associate them with two partitions. We find that the use of partitions is an elegant way to express these polynomials and we restate some of their known properties in terms of partitions. We discuss the asymptotic behavior of the regular zeros and the exceptional zeros of exceptional Laguerre polynomials as the degree tends to infinity.
The linear transformation that sends $x^n$ to the n'th Laguerre polynomial preserves real-rootedness.
We investigate the interlacing of zeros of polynomials of different degrees within the sequences of $q$-Laguerre polynomials $\left\{\tilde{L}_n^{(δ)}(z;q)\right\}_{n=0}^{\infty}$ characterized by $δ\in(-2,-1).$ The interlacing of zeros of quasi-orthogonal polynomials $\tilde{L}_n^{(δ)}(z;q)$ with those of the orthogonal polynomials $\tilde{L}_m^{(δ+t)}(z;q), m,n\in\mathbb{N}, t\in\{1,2\}$ is also considered. New bounds for the least zero of the (order $1$) quasi-orthogonal $q$-Laguerre polynomials are derived.
Let $L_n(x)$ and $L_n^α(x)$ be the $n$th Laguerre and associated Laguerre polynomial respectively. Fisk proved that the linear operator sending $x^n$ to $L_n(x)$ preserves real-rootedness. In this note we prove a stronger result; namely, that when $α\ge 0$, the linear operator sending $x^n$ to $(-1)^n n! L_n^α(x)$ preserves real-rootedness.
In this paper we prove in a constructing way that exceptional Charlier, Meixner, Hermite and Laguerre polynomials satisfy higher order recurrence relations. Our conjecture is that the recurrence relations provided in this paper have minimal order.
This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[ \frac{1}{x-c}x^{α}e^{-x}dx+Nδ(x-c), \] for $x\in[0,\infty)$, $α>-1$, a free parameter $N\in \mathbb{R}_{+}$ and a shift $c<0$. We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as $n$ tends to infinity.
In this paper we consider a semi-classical variation of the weight related to the little $q$-Laguerre polynomials and obtain a second order second degree discrete equation for the recurrence coefficients in the three-term recurrence relation.
It is stated and proved a characterization theorem for Laguerre-Hahn orthogonal polynomials on non-uniform lattices. This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.
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 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…
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.
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.
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。
In this paper we compute the leading terms in the sum of the $k^{th}$ power of the roots of $L_{p}^{(α)}$, the Laguerre-polynomial of degree $p$ with parameter $α$. The connection between the Laguerre-polynomials and the Marchenko-Pastur distribution is expressed by the fact, among others, that the limiting distribution of the empirical distribution of the normalized roots of the Laguerre-polynomials is given by the Marchenko-Pastur distribution. We give a direct proof of this statement based on the recursion satisfied by the Laguerre-polynomials. At the same time, our main result gives that t…
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions.
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…
Confluent functions, Laguerre polynomials and their (generalized) bilinear integrals
本文研究了连函数与拉格朗日多项式的双线性积分性质，揭示了其在不同参数下的正交关系及广义格拉姆矩阵的构建方法。
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…
We propose a simple uniform lower bound on the spacings between the successive zeros of the Laguerre polynomials $L_n^{(α)}$ for all $α>-1$. Our bound is sharp regarding the order of dependency on $n$ and $α$ in various ranges. In particular, we recover the orders given in \cite{ahmed} for $α\in (-1,1]$.
We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure $ρ_{a,c}$ and Hahn measure $ρ_{a,b,N}$ by removing a finite number of their mass points when the parameter $c$ of the Meixner measure and $a$ or $b$ of the Hahn measure are positive integers. From the new Krall-Meixner families we construct new families of exceptional Meixner and Laguerre pol…
New results on the associated Meixner, Charlier, Laguerre, and Krawtchouk polynomials
本文研究四类关联正交多项式，给出其新的显式表示与生成函数，并据此推导对应经典多项式的生成函数、卷积型公式及相关推论。
Expressions for the summation of the series involving the Laguerre polynomials \[S_m(\pmν, \pm p)\equiv e^{-x}\sum_{n=0}^\infty \frac{x^n\,L_n^{(ν)}(x)}{(1\pm ν\pm p)_n}\frac{(f+m)_n}{(f)_n}\] for any non-negative integers $m$ and $p$ are obtained in terms of generalized hypergeometric functions. These results extend previous work in the literature.
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.
In a previous paper we deformed Hermite polynomials to three associated polynomials .Here we apply the same deformation to Laguerre polynomials .
The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an explicit expression, a Rodrigues' type formula and expressions for the derivatives. The novelty of the present paper is that it is the first paper on degenerate versions of orthogonal polynomials.
Orthogonal q-polynomials associated with q-Laguerre-Hahn form will be studied as a generalization of the q-semiclassical forms via a suitable q-difference equation. The concept of class and a criterion to determinate it will be given. The q-Riccati equation satisfied by the corresponding formal Stieltjes series is obtained. Also, the structure relation is established. Some illustrative examples are highlighted.
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…
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…
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)
It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these para…
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。
A two-parameter sequence of orthogonal polynomials $\{P_n( x; λ, t)\}_{n\ge 0}$ with respect to the weight function $x^αe^{- λx} ρ_ν(x t),\ α> -1,\ λ, t \ge 0, \ ρ_ν(x)= 2 x^{ν/2} K_ν(2\sqrt x),\ x >0, ν\ge 0$, where $K_ν(z)$ is the modified Bessel function, is investigated. The case $λ=0$ corresponds to the Prudnikov polynomials and $t=0$ is related to the Laguerre polynomials. A special one-parameter case $\{P_n( x; 1-t, t)\}_{n\ge 0},\ t \in [0,1]$ is analyzed as well.
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.
Using Casorati determinants of Meixner polynomials $(m_n^{a,c})_n$, we construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers a sequence of polynomials $m_n^{a,c;\F}$, $n\in σ_\F$, which are eigenfunctions of a second order difference operator, where $σ_\F$ is certain infinite set of nonnegative integers, $σ_\F \varsubsetneq \NN$. When $c$ and $\F$ satisfy a suitable admissibility condition, we prove that the polynomials $m_n^{a,c;\F}$, $n\in σ_\F$, are actually exceptional Meixner polynomials; that is, in addition, they are orthogonal and complete with respect to a positiv…
The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis an analogous strange orthogonality is obtained for Meixner polynomials.
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.
On the product of the extreme zeros of Laguerre polynomials
本文研究 Laguerre 多项式极端零点乘积的未知性质，并借助参数特征值问题推进求解相关的 Hermite 多项式二十年猜想。
We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely $R_n=L_n^α+aL_{n}^{α'}$ and $S_n=L_n^α+bL_{n-1}^{α'}$. Proofs and numerical counterexamples are given in situations where the zeros of $R_n$, and $S_n$, respectively, interlace (or do not in general) with the zeros of $L_k^α$, $L_k^{α'}$, $k=n$ or $n-1$. The results we prove hold for continuous, as well as integral, shifts of the parameter $α$.
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.
The q-Laguerre polynomials correspond to an indetermined moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's ${}_1ψ_1$-summation we complement the orthogonal q-Laguerre polynomials into an explicit orthogonal basis for the corresponding L^2-space. The dual orthogonal system consists of so-called big q-Bessel functions, which can be obtained as a rigorous limit of the orthogonal system of big q-Jacobi polynomials. Interpretations on the SU(1,1) and E(2) quantum groups are discussed.
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 look at some extensions of the Stieltjes-Wigert weight functions. First we replace the variable x by x^2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal polynomials can be expressed in terms of a solution of the q-discrete Painlevé III equation. Next we consider the q-Laguerre or generalized Stieltjes-Wigert weight functions with a quadratic transformation and derive recursive equations for the recurrence coefficients of the orthogonal polynomials. These turn out to be related to the q-discrete Painlevé V eq…
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