Classical Orthogonal Polynomials Research Articles

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

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.

A New First Finite Class of Classical Orthogonal Polynomials Operational Matrices: An Application for Solving Fractional Differential Equations

In this paper, new operational matrices (OMs) of ordinary and fractional derivatives (FDs) of a first finite class of classical orthogonal polynomials (FFCOP) are introduced. Also, two algorithms are proposed for using the tau and collocation spectral methods (SPMs) to get new approximate solutions to the given fractional differential equations (FDEs). These algorithms convert the given FDEs subject to initial/boundary conditions (I/BCs) into linear or nonlinear systems of algebraic equations that can be solved using appropriate solvers. To demonstrate the robustness, efficiency, and accuracy of the proposed spectral solutions, several illustrative examples are presented. The obtained results show that the proposed algorithms exhibit higher accuracy compared to existing techniques in the literature. Furthermore, an error analysis is provided.

Simultaneous Approximation via Laplacians on the Unit Ball

We study the orthogonal structure on the unit ball textbf{B}^d of mathbb {R}^d with respect to the Sobolev inner products f,gΔ=λL(f,g)+∫BdΔ[(1-‖x‖2)f(x)]Δ[(1-‖x‖2)g(x)]dx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\langle f,g\\right\\rangle _{\\Delta } =\\lambda \\, \\mathscr {L}(f,g) + \\int _{\ extbf{B}^d}{\\Delta [(1-\\Vert x\\Vert ^2) f(x)] \\, \\Delta [(1-\\Vert x\\Vert ^2) g(x)]\\,\ extrm{d}x}, \\end{aligned}$$\\end{document}where mathscr {L}(f,g) = int _{textbf{S}^{d-1}}f(xi ),g(xi ),textrm{d}sigma (xi ) or mathscr {L}(f,g) = f(0) g(0), lambda >0, sigma denotes the surface measure on the unit sphere textbf{S}^{d-1}, and Delta is the usual Laplacian operator. Our main contribution consists in the study of orthogonal polynomials associated with langle cdot , cdot rangle _{Delta }, giving their explicit expression in terms of the classical orthogonal polynomials on the unit ball, and proving that they satisfy a fourth-order partial differential equation, extending the well-known property for ball polynomials, since they satisfy a second-order PDE. We also study the approximation properties of the Fourier sums with respect to these orthogonal polynomials and, in particular, we estimate the error of simultaneous approximation of a function, its partial derivatives, and its Laplacian in the L^2(textbf{B}^d) space.

The Stenger conjectures and the A-stability of collocation Runge-Kutta methods

Stenger conjectures are claims about the location of the eigenvalues of matrices whose elements are certain integrals involving basic Lagrange interpolating polynomials supported on the zeros of orthogonal polynomials. In this paper, we show the validity of the extended Stenger conjecture for families of classical orthogonal polynomials. We also show the validity of the restricted Strenger conjecture for a family of Jacobi and generalized Laguerre orthogonal polynomials. A connection with the A-stability of the collocation Runge-Kutta methods is investigated.

Orthogonal polynomials with respect to the Abel weight

While a lot is known about the classical orthogonal polynomials, their counterparts with respect to non-classical weights are not as well explored. Nevertheless, sometimes such weights come handy as well. For example, the famous Abel–Plana summation formula offers a convenient method of summing an infinite series, reducing the sum to an integral with the Abel weight function on the real line, w(x)=x2sinhπx. Orthogonal polynomials with respect to this weight naturally arise when we have to numerically evaluate this integral using the Gauss quadrature rule. These orthogonal polynomials are the object of study in this paper. We obtain a number of explicit formulas and algebraic relations between these and related polynomials, including the associated polynomials. In particular, for many of these polynomials we obtain Fourier expansions with the orthogonal polynomials as the basis. We also determine the weight functions whose orthogonal polynomials are the polynomials we discussed. At the end, we briefly discuss the asymptotic and perform numerical experiments.

More properties of (β,γ)-Chebyshev functions and points

Recently, (β,γ)-Chebyshev functions, as well as the corresponding zeros, have been introduced as a generalization of classical Chebyshev polynomials of the first kind and related roots. They consist of a family of orthogonal functions on a subset of [−1,1], which indeed satisfies a three-term recurrence formula. In this paper we present further properties, which are proven to comply with various results about classical orthogonal polynomials. In addition, we prove a conjecture concerning the Lebesgue constant's behavior related to the roots of (β,γ)-Chebyshev functions in the corresponding orthogonality interval.

Approximation by polynomials in Sobolev spaces associated with classical moment functionals

Let textbf{u} be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in H^r(textbf{u}), the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L^2 space associated with textbf{u}. This requires the simultaneous approximation of a function f and its consecutive derivatives up to order Nleqslant r. We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of E_n(f^{(r)}), the error of best approximation of f^{(r)} in L^{2}(textbf{u}).

Rationally-extended Dunkl oscillator on the line

It is shown that the extensions of exactly-solvable quantum mechanical problems connected with the replacement of ordinary derivatives by Dunkl ones and with that of classical orthogonal polynomials by exceptional orthogonal ones can be easily combined. For such a purpose, the example of the Dunkl oscillator on the line is considered and three different types of rationally-extended Dunkl oscillators are constructed. The corresponding wavefunctions are expressed in terms of exceptional orthogonal generalized Hermite polynomials, defined in terms of the three different types of X m -Laguerre exceptional orthogonal polynomials. Furthermore, the extended Dunkl oscillator Hamiltonians are shown to be expressible in terms of some extended Dunkl derivatives and some anharmonic oscillator potentials.

Finite elements based on Jacobi shape functions for the free vibration analysis of beams, plates, and shells

This article proposes the use of Jacobi polynomials as shape functions for the free vibration analysis of beam, plate, and shell structures. Jacobi polynomials, indicated as P p ( γ , θ ) , belong to the family of classical orthogonal polynomials, and depend on two scalar parameters γ and θ, with p being the polynomial order. The Jacobi-like shape functions are built in the context of the Carrera unified formulation, which permits the expression of displacement kinematics in a hierarchical form. In this manner, it is possible to adopt several classical to complex higher-order theories with ease. Particular attention is focused on the attenuation and the correction of the shear locking. The results have been compared with analytical results from the literature. For the plate benchmark, analytical results are introduced as the reference results in this article for the first time using the closed form of CUF. Beams, plates, and shells with different thicknesses have been considered. It is demonstrated that the parameters γ and θ are not influential for the calculations.