Fourth-order difference equation for the associated Meixner and Charlier polynomials

An explicit representation of the associated Meixner polynomials (with a real association parameter γ?0) is given in terms of hypergeometric functions. This representation allows to derive the fourth-order difference equation verified by these polynomials. Appropriate limits give the fourth-order difference equation for the associated Charlier polynomials and the fourth-order differential equations for the associated Laguerre and Hermite polynomials.     

Asymptotic Approximations between the Hahn-Type Polynomials and Hermite, Laguerre and Charlier Polynomials

It has been shown in Ferreira et al. (Adv. Appl. Math 31:61–85, [2003]), López and Temme (Methods Appl. Anal. 6:131–196, [1999]; J. Cpmput. Appl. Math. 133:623–633, [2001]) that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions. In this paper we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.      

Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials

It has been shown in Ferreira et al. [Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials, Adv. in Appl. Math. 31(1) (2003) 61–85], López and Temme [Approximations of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999) 131–146; The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001) 623–633] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations. In Ferreira et al. [Limit relations between the Hahn polynomials and the Hermite, Laguerre and Charlier polynomials, submitted for publication] we have established new asymptotic connections between the fourth level and the two lower levels. In this paper, we continue with that program and obtain asymptotic expansions between the fourth level and the third level: we derive 16 asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials. From these expansions, we also derive three new limits between those polynomials. Some numerical experiments show the accuracy of the approximations and, in particular, the accuracy in the approximation of the zeros of those polynomials.     

The dual multivariate Charlier and Edgeworth expansions

The Charlier differential series for distribution and density functions is the foundation for the Edgeworth expansions of distribution and density functions of sample estimators. Here, we give two forms of these expansions for multivariate distributions using multivariate Bell polynomials. Two forms arise because the multivariate Hermite polynomials have a dual form. These dual forms for the multivariate Charlier and Edgeworth expansions appear to be new.     

Coefficients of Wronskian Hermite polynomials

We study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the p-star.     

Generalized and mixed type Gegenbauer polynomials

In this paper, a new generalized form of the Gegenbauer polynomials is introduced by using the integral representation method. Further, the Hermite–Gegenbauer and the Laguerre–Gegenbauer polynomials are introduced by using the operational identities associated with the generalized Hermite and Laguerre polynomials of two variables.     

Monotonicity of the zeros of orthogonal polynomials through related measures

Relation between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), is well known. We use this relation to study the monotonicity properties of the zeros of generalized orthogonal polynomials. As examples, the Jacobi, Laguerre and Charlier polynomials are considered.     

Some characteristic property of the Meixner polynomials

A new characterization of the Meixner polynomials is established. It is based on the solution of a problem related to a previous result concerning the Laguerre polynomials. Solutions of analogous problems provide characteristic properties of the Laguerre and Hermite polynomials. These properties, which are derived from the two-variable polynomials, generalize, in turn, the previous ones.     

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand‐alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15–16 digits) of the nodes and weights of the Gauss–Hermite and Gauss–Laguerre quadratures.     

Higher-order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre polynomials—can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.     

Two new multivariable generating relations

A multidimensional extension of Bailey's transform is utilised to deduce two new generating relations of quite a general character. These expressions are then specialised to give more practical formulae in terms of Karlsson's generalised Kampé de Fériet functions which embody very many generating relations. A number of interesting special cases are given in an appendix which includes results involving Lauricella polynomials, generalised hypergeometric polynomials and the polynomials of Meixner, Charlier and Laguerre.     

Sequence of operations analysis for dynamic data structures

This paper introduces a rather general technique for computing the average-case performance of dynamic data structures, subjected to arbitrary sequences of insert, delete, and search operations. The method allows us effectively to evaluate the integrated cost of various interesting data structure implementations, for stacks, dictionaries, symbol tables, priority queues, and linear lists; it can thus be used as a basis for measuring the efficiency of each proposed implementation. For each data type, a specific continued fraction and a family of orthogonal polynomials are associated with sequences of operations: Tchebycheff for stacks, Laguerre for dictionaries, Charlier for symbol tables, Hermite for priority queues, and Meixner for linear lists. Our main result is an explicit expression, for each of the above data types, of the generating function for integrated costs, as a linear integral transform of the generating functions for individual operation costs. We use the result to compute explicitly integrated costs of various implementations of dictionaries and priority queues.     

Polynomial sequences associated with the classical linear functionals

This work in mainly devoted to the study of polynomial sequences, not necessarily orthogonal, defined by integral powers of certain first order differential operators in deep connection to the classical polynomials of Hermite, Laguerre, Bessel and Jacobi. This connection is streamed from the canonical element of their dual sequences. Meanwhile new Rodrigues-type formulas for the Hermite and Bessel polynomials are achieved.     

Resonant Equations with Classical Orthogonal Polynomials. II

Ukrainian Mathematical Journal - We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials,...     

Finite Differences and Orthogonal Polynomials

By combining finite differences with symmetric functions, we present an elementary demonstration for the limit relation from Laguerre to Hermite polynomials, proposed by Richard Askey. Another limit relation between these two polynomials will also be established.     

Incomplete 2D Hermite polynomials: properties and applications

Incomplete forms of two-variable two-index Hermite polynomials are introduced. Their link with Laguerre polynomials is discussed and it is shown that they are a useful tool to study quantum mechanical harmonic oscillator entangled states. The possibility of developing the theory of complete 2D Hermite polynomials from the point of view of the incomplete forms is analyzed too. The orthogonality properties of the associated harmonic-oscillator functions are finally discussed.     

Two new multivariable generating relations

A multidimensional extension of Bailey's transform is utilised to deduce two new generating relations of quite a general character. These expressions are then specialised to give more practical formulae in terms of Karlsson's generalised Kampé de Fériet functions which embody very many generating relations. A number of interesting special cases are given in an appendix which includes results involving Lauricella polynomials, generalised hypergeometric polynomials and the polynomials of Meixner, Charlier and Laguerre.     

Generalized polynomials and new families of generating functions

We show that the use of generalized polynomials, i.e. polynomials defined as discrete convolution of Hermite, Laguerre… polynomials can be exploited to explore new families of generating functions.

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Sunto Si dimostra che l'uso di polinomi generalizzati definiti come convoluzioni discrete di polinomi di Hermite, Laguerre etc. possono essere utilizzati per studiare nuove famiglie di funzioni generatrici.

     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

Sub-exponentially localized kernels and frames induced by orthogonal expansions

The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions.