Classical Orthogonal Polynomials Research Articles

On the functional equation for classical orthogonal polynomials on lattices

Necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials on lattices are stated. Moreover, the functional Rodrigues formula and a closed formula for the recurrence coefficients are presented.

An urn model for the Jacobi-Piñeiro polynomials

The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T. Ehrenfest and eventually connected with the Krawtchouk and Hahn polynomials. This connection was reversed recently in the case of the Jacobi polynomials where a rather contrived, and later a simpler urn model was proposed. Here we consider an urn model going with the Jacobi-Piñeiro multiple orthogonal polynomials. These polynomials have recently been put forth in connection with a stochastic matrix.

Moments of orthogonal polynomials and exponential generating functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( $$q=1$$ ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.

Numerical solution of time-fractional telegraph equation by using a new class of orthogonal polynomials

‎In this article‎, ‎an efficient numerical method based on a new class of orthogonal polynomials‎, ‎namely Chelyshkov polynomials‎, ‎has been presented to approximate solution of time-fractional telegraph (TFT) equations‎. ‎The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations‎. ‎The error analysis of the proposed collocation method is also investigated‎. ‎A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations‎. ‎Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method‎.

Application of the discrete classical case to a 1−2 type relation

In this paper, we present a simple approach in order to build up recursively the connection coefficients between a sequence of polynomials {Q n } n≥0 and an orthogonal polynomials sequence {P n } n≥0 when P n (x) = Q n (x) +r n Q n −1 (x), n≥0. This yields the relation between the parameters of the corresponding recurrence relations. Some special cases are developed. More specifically, assuming that {P n } n≥0 is a discrete classical orthogonal polynomials sequence.

Riordan posets and associated incidence matrices

This paper introduces a new class of partially ordered sets represented as binary Riordan matrices referred to as ‘Riordan posets’. This notion extends the theory of Riordan matrices into the domain of poset theory. We establish the criterion for a given binary Riordan matrix to be defined as a Riordan poset matrix. It is also shown that every Riordan poset is a locally finite poset. This leads to the construction of various matrix algebras obtained from incidence algebras of Riordan posets. Many structural properties of Riordan posets are studied and various families of Riordan posets are introduced. A class of series-parallel posets is derived by extending the notion of Riordan posets to include exponential Riordan matrices, and it is obtained from Sheffer sequences of classical orthogonal polynomials.

Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

AbstractWe construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second‐order difference or differential operator. In mathematical physics, they allow the explicit computation of bound states of rational extensions of classical quantum‐mechanical potentials. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

Fast associated classical orthogonal polynomial transforms

We discuss a fast approximate solution to the associated classical–classical orthogonal polynomial connection problem. We first show that associated classical orthogonal polynomials are solutions to a fourth-order quadratic eigenvalue problem with polynomial coefficients such that the differential operator is degree-preserving. Upon linearization, the discretization of this quadratic eigenvalue problem is block upper-triangular and banded. After a perfect shuffle, we extend a divide-and-conquer approach to the upper-triangular and banded generalized eigenvalue problem to the blocked case, which may be accelerated by one of a few different algorithms. Associated orthogonal polynomials arise from iterated Stieltjes transforms of orthogonal polynomials; hence, fast approximate conversion to classical cases combined with fast discrete sine and cosine transforms provides a modular mechanism for synthesis of singular integral transforms of classical orthogonal polynomial expansions.

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Given a linear second-order differential operator {mathcal {L}}equiv phi ,D^2+psi ,D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers lambda _n, ngeqslant 0, and a Sobolev bilinear form B(p,q)=∑k=0Nuk,p(k)q(k),N⩾0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {B}}(p,q)\\,=\\,\\sum _{k=0}^N\\left\\langle {{\\mathbf {u}}_k,\\,p^{(k)}\\,q^{(k)}}\\right\\rangle , \\quad N\\geqslant 0, \\end{aligned}$$\\end{document}where {mathbf {u}}_k, 0leqslant k leqslant N, are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation {mathcal {L}}[y]=lambda _n,y with respect to {mathcal {B}}. We show that such polynomials are orthogonal with respect to {mathcal {B}} if the Pearson equations D(phi ,{mathbf {u}}_k)=(psi +k,phi '),{mathbf {u}}_k, 0leqslant k leqslant N, are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

Stable Calculation of Krawtchouk Functions from Triplet Relations

Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples.

Gibbs phenomena for some classical orthogonal polynomials

The Gibbs phenomena associated to partial sums of Fourier series are now well understood. In this paper, we show that Gibbs phenomena also occur for expansions of functions in terms of members of one of several general classes of orthogonal polynomials, in particular treating in a relatively unified manner expansions in either Legendre (more generally Jacobi), Hermite, or Laguerre polynomials. Noteworthy is the fact that the Gibbs constants associated to all of these expansions have the same value, approximately 0.0893 or more precisely 1π∫0πsin⁡ttdt−12.

Positivity of Turán determinants for orthogonal polynomials II

The polynomials pn orthogonal on the interval [−1,1], normalized by pn(1)=1, satisfy Turán’s inequality if pn2(x)−pn−1(x)pn+1(x)≥0 for n≥1 and for all x in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Turán’s inequality. This extends essentially the results of Szwarc(1998). In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.

On Three Families of Karhunen–Loève Expansions Associated with Classical Orthogonal Polynomials

On Three Families of Karhunen–Loève Expansions Associated with Classical Orthogonal Polynomials

Generalized coherent states of exceptional Scarf-I potential: Their spatio-temporal and statistical properties

We construct generalized coherent states for the rationally extended Scarf-I potential. Statistical and geometrical properties of these states are investigated. Special emphasis is given to the study of spatio-temporal properties of the coherent states via the quantum carpet structure and the auto-correlation function. Through this study, we aim to find the signature of the “rationalization” of the conventional potentials and the classical orthogonal polynomials.

Christoffel transform of classical discrete measures and invariance of determinants of classical and classical discrete polynomials

Given a finite set of complex numbers U and a classical discrete measure μ, we consider the Christoffel transform ∏u∈U(x−u)μ. Under mild conditions on the finite set U, we show that the orthogonal polynomials with respect to ∏u∈U(x−u)μ enjoy a bunch of non trivial determinantal representations in terms of the classical discrete orthogonal polynomials with respect to μ. Under additional assumptions on the finite set U, we dualize these determinantal representations and find some invariance properties for quasi Casoration determinants whose entries are classical discrete orthogonal polynomials. Passing to the limit we recover some known invariance properties for Wronskian determinant whose entries are classical polynomials.

Computational aspects of fractional Romanovski–Bessel functions

Nowadays, various groups of classical orthogonal polynomials are part of several spectral algorithms for basic mathematical machinery. It is well known that the polynomial-based spectral methods can provide high accuracy for partial differential equations with smooth solutions, but may not have any advantage when the solutions exhibit weakly singular behavior. However, to establish accurate spectral schemes for problems with non-smooth solutions, one has to enrich the polynomial-based approximation space by introducing special functions that capture the singular behavior of the underlying problem. The main purpose of this paper is to introduce a new finite class of orthogonal functions based on the Romanovski–Bessel polynomials, and to investigate their basic general properties such as the fractional Romanovski–Bessel–Gauss-type quadrature formulae together with fundamental results of approximation for certain weighted projection operators for certain weighted projection operators described in suitable weighted Sobolev spaces. The relationship between such new orthogonal finite set of functions and the other families of infinite fractional orthogonal functions such as fractional Laguerre functions and generalized Laguerre fractional modified functions are derived.

Algorithms for Construction of Recurrence Relations for the Coefficients of the Fourier Series Expansions with Respect to Classical Discrete Orthogonal Polynomials

A new formula expressing explicitly the integrals, antidifference, of discrete orthogonal polynomials $$\{P_{n}(x):$$ Hahn, Meixner, Kravchuk, and Charlier $$\}$$ of any degree in terms of $$P_{n}(x)$$ themselves are proved. Other formulae for the expansion coefficients of general-order difference integrations $$\nabla ^{-s}f(x),\,\Delta ^{-s}f(x),$$ $$\nabla ^{-s}[x^{\ell }\nabla ^{q}f(x)]\,$$ and $$\Delta ^{-s}[x^{\ell }\Delta ^{q}f(x)],$$ of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained.

$q$-Deformation of the Kac-Sylvester tridiagonal matrix

Upon replacing integers by $q$-integers, one arrives at a natural $q$-deformation of a remarkably ubiquitous tridiagonal integer matrix whose eigenvalues and eigenvectors were first computed by J.J. Sylvester and M. Kac, respectively. The present note computes the eigenvalues and eigenvectors of this novel $q$-deformed Kac-Sylvester matrix through a connection with Macdonald’s hyperoctahedral Hall-Littlewood polynomials. This yields the spectrum as a minimizer of an associated Morse function reminiscent of Stieltjes’ electrostatic potential for the roots of classical orthogonal polynomials.

Robust Stabilization of Interval Plants with Uncertain Time-Delay Using the Value Set Concept

This paper considers the robust stabilization problem for interval plants with parametric uncertainty and uncertain time-delay based on the value set characterization of closed-loop control systems and the zero exclusion principle. Using Kharitonov’s polynomials, it is possible to establish a sufficient condition to guarantee the robust stability property. This condition allows us to solve the control synthesis problem using conditions similar to those established in the loopshaping technique and to parameterize the controllers using stable polynomials constructed from classical orthogonal polynomials.

Palindromic Riordan arrays, classical orthogonal polynomials and Catalan triangles

We characterize the palindromic generalized Riordan arrays and their Sheffer sequences showing that, apart from a trivial case, these arrays all arise from a given array P(q,t,u) by suitably choosing the parameters q,t,u. Remarkably, well-known polynomial and numerical sequences arise as special cases of such Sheffer sequences, including classical orthogonal polynomials. After a suitable normalization, both the palindromic Sheffer sequences and their gamma polynomials have non-negative integer coefficients. We prove that these coefficients count a family of directed graphs, then through this combinatorial setting we obtain a bijective proof of the palindromic property of P(q,t,u), and we recover the combinatorial expansion of orthogonal polynomials due to F. Bergeron. Finally, we state explicit connections between the inverse Q(q,t,u) of P(q,t,u) and an array C(q,t) which generalizes Aigner's array of ballot numbers, the Pascal triangle and the Catalan triangle of Shapiro.