Discrete 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.

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.

GICAR Algebras and Dynamics on Determinantal Point Processes: Discrete Orthogonal Polynomial Ensemble Case

GICAR Algebras and Dynamics on Determinantal Point Processes: Discrete Orthogonal Polynomial Ensemble Case

Toda and Laguerre–Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.

Matrix factorizations for the generalized Charlier and Meixner orthogonal polynomials

The Cholesky factorization of the moment matrix is considered for the generalized Charlier and generalized Meixner discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre–Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient γn, that happen to be forms of the Painlevé deg-PV in disguise. New Laguerre–Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche.

Laguerre–Freud Equations for the Gauss Hypergeometric Discrete Orthogonal Polynomials

The Cholesky factorization of the moment matrix is considered for the Gauss hypergeometric discrete orthogonal polynomials. This family of discrete orthogonal polynomials has a weight with first moment given by ρ0=2F1a,bc+1;η. For the Gauss hypergeometric discrete orthogonal polynomials, also known as generalized Hahn of type I, Laguerre–Freud equations are found, and the differences with the equations found by Dominici and by Filipuk and Van Assche are provided.

On Two Families of Discrete Multiple Orthogonal Polynomials on a Star-like Set

On Two Families of Discrete Multiple Orthogonal Polynomials on a Star-like Set

Asymptotics of discrete q-Freud orthogonal polynomials from the q-Riemann Hilbert problem

We investigate a Riemann–Hilbert problem (RHP), whose solution corresponds to a group of q-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the polynomials approach infinity. The RHP formulation also enables us to obtain further properties. In particular, we consider how the class of polynomials and their asymptotic behaviours change under translations of the q-discrete lattice and determine the asymptotics of a related q-Painlevé equation.

Laguerre–Freud equations for three families of hypergeometric discrete orthogonal polynomials

AbstractThe Cholesky factorization of the moment matrix is considered for discrete orthogonal polynomials of hypergeometric type. We derive the Laguerre–Freud equations when the first moments of the weights are given by the 1F2, 2F2, and 3F2 generalized hypergeometric series.

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

This paper investigates the new model order reduction (MOR) methods via bivariate discrete orthogonal polynomials for two-dimensional (2-D) discrete systems. The 2-D discrete system is described by the Kurek model. First, we deduce algebraically the shift-transformation matrix of the classical discrete orthogonal polynomials of one variable. By means of the shift-transformation matrices, 2-D discrete systems are expanded in the spaces spanned by bivariate discrete orthogonal polynomials. The coefficient matrices are calculated from matrix equations. Then the reduced-order systems are produced by the orthogonal projection matrices defined by the coefficient matrices. Theoretical analysis shows that the reduced-order systems can match a certain number of coefficient vectors of the original outputs. Finally, one numerical example is simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Numerical solution of delay Volterra functional integral equations with variable bounds

This work presents a new numerical method for solving Volterra-type integro functional equations with variable bounds and mixed delay. This paper applies discrete orthogonal Hahn polynomials and their properties numerically. A discrete scalar product is associated with discrete orthogonal polynomials. Several numerical experiments (including linear and nonlinear) for multiple test problems are provided to validate the accuracy of this method.

Four-Term Recurrence for Fast Krawtchouk Moments Using Clenshaw Algorithm

Krawtchouk polynomials (KPs) are discrete orthogonal polynomials associated with the Gauss hypergeometric functions. These polynomials and their generated moments in 1D or 2D formats play an important role in information and coding theories, signal and image processing tools, image watermarking, and pattern recognition. In this paper, we introduce a new four-term recurrence relation to compute KPs compared to their ordinary recursions (three-term) and analyse the proposed algorithm speed. Moreover, we use Clenshaw’s technique to accelerate the computation procedure of the Krawtchouk moments (KMs) using a fast digital filter structure to generate a lattice network for KPs calculation. The proposed method confirms the stability of KPs computation for higher orders and their signal reconstruction capabilities as well. The results show that the KMs calculation using the proposed combined method based on a four-term recursion and Clenshaw’s technique is reliable and fast compared to the existing recursions and fast KMs algorithms.

Neutron Multiplicity Counting Distribution Reconstruction from Moments Using Meixner Polynomial Expansion and N-Forked Branching Approximations

Methods used to infer nuclear parameters from neutron count statistics fall into two categories depending on whether they use moments or count number probabilities. As probabilities are in general more difficult to calculate, we are interested here in the reconstruction of distributions from their lower-order moments. For this, we explore two approaches. The first one relies on a generalization of the two-forked branching correlation (quadratic) approximation used in the PMZBB and Poisson radical distributions, and the second one is founded on the expansion of the distribution on a Meixner discrete orthogonal polynomial base.

Parallel model order reduction methods based on structured matrix analysis for discrete-time systems with parametric uncertainty

This paper explores two novel parallel parametric model order reduction methods for discrete-time systems with parametric uncertainty. The proposed methods allow that the parametric dependence is non-affine. With the shift-transformation matrix of the discrete orthogonal polynomials, including Charlier polynomials and Krawtchouk polynomials, the expansion coefficients of state variable in discrete orthogonal polynomials space can be obtained by solving the system of linear algebraic equations (SLAE). By applying the structured matrix analysis of the SLAE, two parallel strategies based on the equivalent transformation of block bi-diagonal matrices and the block discrete Fourier transform of block ϵ-circulant matrices are proposed to solve the SLAE. Then, the projection matrix is constructed to reduce discrete-time parametric systems. Moreover, we analyze the properties of the proposed parallel methods, including the invariable coefficients, the invertibility, and the error analysis. Finally, the effectiveness of the proposed methods is illustrated by the numerical experiments.

The electrostatic equilibrium problem for classical discrete orthogonal polynomials

We present two types of generic difference characterizations of zeros of the classical discrete orthogonal polynomials associated with a discrete weight function satisfying a difference equation of the Pearson type. These characterizations use either the total forward or modified partial central difference operators of discrete discriminants, which approximate the total electrostatic energy of the system of n n unit charges moving freely on an interval in presence of external potential given by the iterated discrete weight.

Reduced-order modeling methods via bivariate discrete orthogonal polynomials for two-dimensional discrete state-delayed systems

Reduced-order modeling methods via bivariate discrete orthogonal polynomials for two-dimensional discrete state-delayed systems

Comparative asymptotics for discrete semiclassical orthogonal polynomials

We study the ratio [Formula: see text] asymptotically as [Formula: see text], where the polynomials [Formula: see text] are orthogonal with respect to a discrete linear functional and [Formula: see text] denote the falling factorial polynomials. We give recurrences that allow the computation of high order asymptotic expansions of [Formula: see text] and give examples for most discrete semiclassical polynomials of class [Formula: see text]. We show several plots illustrating the accuracy of our results.

Topology Optimization Design of an Active Deformable Mirror Based on Discrete Orthogonal Zernike Polynomials

In order to design an active deformation mirror for projection objective aberration imaging quality control, a topology optimization design method of active deformation mirrors based on discrete orthogonal Zernike polynomials is proposed in this paper. Firstly, in order to solve the problem that continuous Zernike polynomials do not have orthogonality on the discrete coordinates inside the unit circle, which causes the instability of topology optimization results, discrete orthogonal Zernike polynomials are used to characterize the active deformation mirror wave aberrations. Then, the optical and structural deformations are combined to establish an optical-mechanical coupling topology optimization model with the help of the variable density method to derive the sensitivity of the mathematical model. Finally, a wave aberration corrected deformation mirror in an optical machine system is used as an arithmetic example for topology optimization, and the results show that the absolute value of the Zernike coefficient Z4 after optimization is improved by nearly one order of magnitude compared with the value before optimization, and the vibration characteristics of the optimized structure meet the design requirements. The optimization effect is significant, which improves the optical performance of the deformed mirror and provides a new scheme for the design of the deformed mirror structure which has a certain practical value for engineering.

Difference equation satisfied by the Stieltjes transform of a sequence

We study a class of generating functions related to the Stieltjes transform of a sequence of moments with respect to the basis of falling factorial polynomials. Given a recurrence relation for the coefficient sequence, it is shown how to compute the difference equation satisified by its generating function w.r.t. this basis. We give several examples from the class of discrete semiclassical orthogonal polynomials.

Basic theory of finite-type relations between orthogonal polynomial sequences

A theory of finite-type relations between two orthogonal polynomial sequences is developed. In fact, we find necessary and sufficient conditions for the extended connection coefficients which should be fulfilled for a finite-type relation. Next, a simplification criterion is given and some applications in the domain of continuous, discrete and quantum semiclassical orthogonal polynomials are treated into detail.