Classical Orthogonal Polynomials Research Articles

Asymptotics of solutions of special second-order linear recurrencies with polynomial coefficients and boundary effects of polynomial filters

In this paper we prove that classical discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights) of high degrees have extremely small values near the endpoints (we call this property “rapid decay near the endpoints”) but extremely large values between these grid points and their roots are very close to the grid points near the endpoints. These results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”).Our results give interesting examples of nontrivial asymptotics of practically important solutions of special second-order linear recurrencies with polynomial coefficients studied by M.Petkovšek; to his memory we dedicate this paper.

Numerical analysis of fractional‐order Euler–Bernoulli beam model under composite model

The primary objective of this study is to develop a new constitutive model by combining a fractional‐order Kelvin–Voigt model with an Abel dashpot element in parallel. Subsequently, this new model will be incorporated into the Euler–Bernoulli beam's governing equation, utilizing shifted Legendre polynomials as basis functions, a classical orthogonal polynomial system, to solve the fractional‐order partial differential equations. By comparing the numerical solutions with the analytical solutions, we aim to evaluate the applicability of shifted Legendre polynomials in solving such problems and the accuracy of the obtained numerical solutions. Furthermore, we will investigate the performance of viscoelastic HDPE beams under different loading conditions and conduct a comparative analysis of the displacements of HDPE beams under the new constitutive model and the traditional fractional‐order Kelvin–Voigt model. Through this research, we hope to gain a deeper understanding of the characteristics of fractional‐order phenomena and provide more accurate and efficient numerical simulation and analysis methods for the field of structural mechanics, promoting the development of related engineering applications.

Polynomial and Rational Measure Modifications of Orthogonal Polynomials via Infinite-Dimensional Banded Matrix Factorizations

AbstractWe describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials. We present several applications and numerical experiments using an open source implementation which make direct use of these results.

Continued Fractions, Orthogonal Polynomials and Dirichlet Series

Using an experimental mathematics approach, new relations are obtained between Dirichlet-like series for certain periodic coefficients and the moments of certain families of orthogonal polynomials. In addition to the classical hypergeometric orthogonal polynomials, of Racah type and continuous dual Hahn type, a new similar family of orthogonal polynomials intervenes.

On the differential equations of frozen Calogero-Moser-Sutherland particle models

Multivariate Bessel and Jacobi processes describe Calogero-Moser-Sutherland particle models. They depend on a parameter k and are related to time-dependent classical random matrix models like Dysom Brownian motions, where k has the interpretation of an inverse temperature. There are several stochastic limit theorems for k→∞ were the limits depend on the solutions of associated ordinary differential equations (ODEs) where these ODEs admit particular simple solutions which are connected with the zeros of the classical orthogonal polynomials. In this paper we show that these solutions attract all solutions. Moreover we present a connection between the solutions of these multivariate ODEs with associated one-dimensional inverse heat equations. These inverse heat equations are used to compute the expectations of some determinantal formulas for the multivariate Bessel and Jacobi processes.

Low rank approximation in the computation of first kind integral equations with TauToolbox

Tau Toolbox is a mathematical library for the solution of integro-differential problems, based on the spectral Lanczos' Tau method. Over the past few years, a class within the library, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.In this work we discuss how this class has been designed to solve linear ill-posed problems and we provide a description of the available methods, Tikhonov regularization and truncated singular value expansion. For the solution of the Fredholm integral equation of the first kind, which is built from a low-rank approximation of the kernel followed by a numerical truncated singular value expansion, an error estimate is given.Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems are used to evaluate the performance and reliability of the solvers.The final product of this paper is the numerical solution of a first-kind integral equation, which is constructed using only two inputs from the user: the kernel and the right-hand side.

A Matlab package computing simultaneous Gaussian quadrature rules for multiple orthogonal polynomials

The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials.Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to r different measures, with r≥1. Moreover, they satisfy (r+2)-term recurrence relations. In this manuscript, without loss of generality, r is considered equal to 2. The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the Matlab function balance.m does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a Matlab package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem.

Sturm’s Comparison Theorem for Classical Discrete Orthogonal Polynomials

In an earlier work (Castillo et al. in J Math Phys 61:103505, 2020), it was established, from a hypergeometric-type difference equation, tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In this work, we continue with the study of zeros of these polynomials by giving a comparison theorem of Sturm type. As an application, we analyze in a simple way some relations between the zeros of certain classical discrete orthogonal polynomials.

Diverse characterizations of Q-Dunkl classical orthogonal polynomials

In this paper, we present four characterizations of q-Dunkl-classical orthogonal polynomials. The first one is a T θ , q -distributional equation of Pearson type fulfilled by its associated form, where T θ , q is the q-Dunkl operator. The second is a second order linear T θ , q -difference equation. The third is a first order linear T θ , q -difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The fourth is the so-called structure relation fulfilled by q-Dunkl-classical polynomials. Moreover, we provide an example of a non-symmetric sequence of T θ , q -classical orthogonal polynomials.

Connecting exceptional orthogonal polynomials of different kind

The known asymptotic relations interconnecting Jacobi, Laguerre, and Hermite classical orthogonal polynomials are generalized to the corresponding exceptional orthogonal polynomials of codimension m. It is proved that Xm-Laguerre exceptional orthogonal polynomials of type I, II, or III can be obtained as limits of Xm-Jacobi exceptional orthogonal polynomials of the same type. Similarly, Xm-Hermite exceptional orthogonal polynomials of type III can be derived from Xm-Jacobi or Xm-Laguerre ones. The quadratic transformations expressing Hermite classical orthogonal polynomials in terms of Laguerre ones is also extended to even X2m-Hermite exceptional orthogonal polynomials.

Subdomain collocation method based on successive integration technique for solving delay differential equations

The main objective of this work is to propose the polynomial based Subdomain collocation method using successive integration technique for solving delay differential equations (DDEs). In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear DDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear DDEs in real-world problems.

Solving neutral delay differential equations using least square method based on successive integration technique

The main objective of this work is to propose the Least square method (LSM) using successive integration technique for solving Neutral delay differential equations (NDDEs). Continuous LSM and Discrete LSM have been presented by adopting different orthogonal polynomials as weighted basis functions. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear NDDEs have been provided to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems.

The Stieltjes–Fekete Problem and Degenerate Orthogonal Polynomials

Abstract A result of Stieltjes famously relates the zeroes of the classical orthogonal polynomials with the configurations of points on the line that minimize a suitable energy with logarithmic interactions under an external field. The optimal configuration satisfies an algebraic set of equations: we call this set of algebraic equations the Stieltjes–Fekete problem. In this work we consider the Stieltjes-Fekete problem when the derivative of the external field is an arbitrary rational complex function. We show that, under assumption of genericity, its solutions are in one-to-one correspondence with the zeroes of certain non-hermitian orthogonal polynomials that satisfy an excess of orthogonality conditions and are thus termed “degenerate”. When the differential of the external field on the Riemann sphere is of degree $3$ our result reproduces Stieltjes’ original result and provides its direct generalization for higher degree after more than a century since the original result.

Solving Neutral Delay Differential Equations Using Galerkin Weighted Residual Method Based on Successive Integration Technique and its Residual Error Correction

Objectives: The main objectives of this work are to solve Neutral Delay Differential Equations (NDDEs) using Galerkin weighted residual method based on successive integration technique and to obtain the Estimation of Error using Residual function. Methods: The Galerkin weighted residual method based on successive integration technique is proposed to obtain approximate solutions of the NDDEs. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Fibonacci polynomials are considered. Findings: Numerical examples of linear and nonlinear NDDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. Novelty: From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems. Keywords: Galerkin Weighted Residual method, Polynomials, Hermite, Bernoulli, Chebyshev, Fibonacci, Successive integration technique, Neutral Delay Differential Equations

CO2-brine interfacial tension correlation based on the classical orthogonal polynomials: monovalent salts with common anion

CO2-brine interfacial tension correlation based on the classical orthogonal polynomials: monovalent salts with common anion

Рациональные коэффициенты ортогональных разложений некоторых функций

Expansions of many elementary and special functions in series of orthogonal polynomials have coefficients known explicitly. However, almost always these coefficients are irrational. Thus any numerical method would give these coefficients approximately with computations in any arithmetic. This is also true for the spectral methods, which give effective approximations for holonomic functions. However, in some exceptional cases the coefficients of expansions obtained with spectral methods turn out to be rational, and are computed exactly in rational arithmetic. We consider these expansions in series of some classical orthogonal polynomials. We demonstrate that, in this way, an infinite number of linear forms can be obtained for some irrationalities, in particular, for the Euler constant.

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Hankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of 14 sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.

A Unified Approach to Computing the Zeros of Orthogonal Polynomials

We present a unified approach to calculating the zeros of the classical orthogonal polynomials and discuss the electrostatic interpretation and its connection to the energy minimization problem. This approach works for the generalized Bessel polynomials, including the normalized reversed variant, as well as the Viet\'e--Pell and Viet\'e--Pell--Lucas polynomials. We briefly discuss the electrostatic interpretation for each aforesaid case and some recent advances. We provide zeros and error estimates for various cases of the Jacobi, Hermite, and Laguerre polynomials and offer a brief discussion of how the method was implemented symbolically and numerically with Maple. In conclusion, we provide possible avenues for future research.

An asymptotic development of the Poisson integral for Laguerre polynomial expansions

The purpose of this paper is the study of the rate of convergence of Poisson integrals for Laguerre expansions. The convergence of partial sums of Fourier series of functions in Lp spaces was studied, for several classes of orthogonal polynomials. In the Laguerre case Askey and Waigner proved convergence for functions f∈Lp0,+∞ with 4/3<p<4. In this paper we deal with the Poisson integral Arf0<r<1 which arises by applying Abel’s summation method to the Laguerre expansion of the function f. About 50 years ago, Muckenhoupt intensively studied the Poisson integral for the Laguerre and Hermite polynomials. Among other things he proved pointwise convergence, the convergence by norm, and that the Poisson integral is a contraction mapping in Lp0,∞. Toczek and Wachnicki gave a Voronovskaja-type theorem by calculating the limit 1−r−1Arfx−fx as r→1−, provided that f′′x exists. We generalize this formula by deriving a complete asymptotic development. All its coefficients are explicitly given in a concise form. As an application we apply extrapolation methods in order to improve the rate of convergence of Arfx as r→1−.

Solving Fractional Gas Dynamics Equation Using Müntz–Legendre Polynomials

To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method’s effectiveness and accuracy.