Sequence Of Orthogonal Polynomials Research Articles

On the spectrum of tridiagonal operators in the context of orthogonal polynomials

AbstractThe basis for our studies is a large class of orthogonal polynomial sequences $$(P_n)_{n\in {{\mathbb {N}}}_0}$$ ( P n ) n ∈ N 0 , which is normalized by $$P_n(x_0)=1$$ P n ( x 0 ) = 1 for all $$n\in {\mathbb {N}}_0$$ n ∈ N 0 where the coefficients in the three-term recurrence relation are bounded. The goal is to check if $$x_0 \in {\mathbb {R}}$$ x 0 ∈ R is in the support of the orthogonalization measure $$\mu $$ μ . For this purpose, we use, among other things, a result of G. H. Hardy concerning Cesàro operators on weighted $$l^2$$ l 2 -spaces. These investigations generalize ideas from Lasser et al. (Arch Math 100:289–299, 2013).

On an orthogonal polynomial sequence and its recurrence coefficients

On an orthogonal polynomial sequence and its recurrence coefficients

Asymptotic for Orthogonal Polynomials with Respect to a Rational Modification of a Measure Supported on the Semi-Axis

Given a sequence of orthogonal polynomials {Ln}n=0∞, orthogonal with respect to a positive Borel ν measure supported on R+, let {Qn}n=0∞ be the the sequence of orthogonal polynomials with respect to the modified measure r(x)dν(x), where r is certain rational function. This work is devoted to the proof of the relative asymptotic formula Qn(d)(z)Ln(d)(z)⇉n∏k=1N1ak+iz+akAk∏j=1N2z+bjbj+iBj, on compact subsets of C∖R+, where ak and bj are the zeros and poles of r, and the Ak, Bj are their respective multiplicities.

On self-adjoint operators generated by the fourth-order Laguerre type differential expression

In this paper we address the problem of finding all self-adjoint operators generated by a Lagrangian symmetrizable differential expression in a Hilbert space with a mixed Sobolev inner product strictly discrete on the derivatives. We start giving survey on the initial results in the literature: the Glazman-Krein-Naimark (GKN) theory, the later developed GKN-EM theory as an extension of the first, and the works of Littlejohn and Wellman on the subject.We apply those to the fourth-order Laguerre type differential expression known for having a sequence of orthogonal polynomials as eigenfunctions and a symmetrizable differential expression. As a result, we give a complete description of the operator space defined from the GKN-EM theory, and also a characterization of all possible self-adjoint operators in the Sobolev space where the operator is defined. We validate these results with known examples previously studied in the literature.

Lax-type pairs in the theory of bivariate orthogonal polynomials

Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyze a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes function, moments, a differential equation for the weight, and the bidimensional Toda-type systems.

Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials.

Асимптотика Сеге для системы Анжелеско

We discuss opportunity for extension the strong asymptotics (Szego asymptotics) for more wide class of the sequences of multiple orthogonal polynomials with respect to the system of weights supported by the non-intersecting intervals (Angelesco system). The interest for limiting properties of these polynomials, when their multi-index tends monotonically to in_nity (along of the edges of p-dimensional lattice) has appeared recently, in onnection with spectral problems for the discrete Schrodinger operator (multidimensional Jacobi matrix) on the graph-trees.

Properties and Convergence Analysis of Orthogonal Polynomials, Reproducing Kernels, and Bases in Hilbert Spaces Associated with Norm-Attainable Operators

This research paper delves into the properties and convergence behaviors of various sequences of orthogonal polynomials, reproducing kernels, and bases within Hilbert spaces governed by norm-attainable operators. Through rigorous analysis, the study establishes the completeness of the sequences of monic orthogonal polynomials and orthonormal polynomials, highlighting their comprehensive representation and approximation capabilities in the Hilbert space. The paper also demonstrates the completeness and density attributes of the sequence of normalized reproducing kernels, showcasing its effective role in capturing the intrinsic structure of the space. Additionally, the research investigates the uniform convergence of these sequences, revealing their convergence to essential operators within the Hilbert space. Ultimately, these results contribute to both theoretical understanding and practical applications in various fields by providing insights into function approximation and representation within this mathematical framework.

Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials

Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given.

Some New Families of Finite Orthogonal Polynomials in Two Variables

In this paper, we generalize the study of finite sequences of orthogonal polynomials from one to two variables. In doing so, twenty three new classes of bivariate finite orthogonal polynomials are presented, obtained from the product of a finite and an infinite family of univariate orthogonal polynomials. For these new classes of bivariate finite orthogonal polynomials, we present a bivariate weight function, the domain of orthogonality, the orthogonality relation, the recurrence relations, the second-order partial differential equations, the generating functions, as well as the parameter derivatives. The limit relations among these families are also presented in Labelle’s flavor.

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

A spectral Favard theorem is derived for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The rich knowledge concerning the spectral and factorization properties of oscillatory matrices forms the basis of this theorem, which is formulated in terms of sequences of multiple orthogonal polynomials of types I and II, associated with a set of positive Lebesgue-Stieltjes measures. Additionally, a multiple Gauss quadrature is established, and the corresponding degrees of precision are determined. The spectral Favard theorem finds application in the context of Markov chains with transition matrices having (p + 2) diagonals, extending beyond the birth and death scenario, while still maintaining a positive stochastic bidiagonal factorization. In the finite case, the Karlin–McGregor spectral representation is provided, along with the demonstration of recurrent behavior and explicit expressions for the stationary distributions in terms of the orthogonal polynomials. Analogous results are obtained for countably infinite Markov chains. In this case, the Markov chain may not be recurrent, and its characterization is expressed in relation to the first measure. The ergodicity of the Markov chain is explored, taking into consideration the presence of a mass at 1, which corresponds to eigenvalues associated with the right and left eigenvectors.

Sobolev orthogonal polynomials and spectral methods in boundary value problems

In the variational formulation of a boundary value problem for the harmonic oscillator, Sobolev inner products appear in a natural way. First, we study the sequences of Sobolev orthogonal polynomials with respect to such an inner product. Second, their representations in terms of a sequence of Gegenbauer polynomials are deduced as well as an algorithm to generate them in a recursive way is stated. The outer relative asymptotics between the Sobolev orthogonal polynomials and classical Legendre polynomials is obtained. Next we analyze the solution of the boundary value problem in terms of a Fourier-Sobolev projector. Finally, we provide numerical tests concerning the reliability and accuracy of the Sobolev spectral method.

On the Hahn's property for matrix orthogonal polynomials on the unit circle

In this contribution, we characterize the sequences of matrix orthogonal polynomials on the unit circle (MOPUC) whose derivatives are also orthogonal, i.e. the orthogonal sequences satisfying the Hahn's property. Equivalently, we characterize classical matrix linear functionals on the unit circle.

Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials

Our work studies sequences of orthogonal polynomials of the Laguerre–Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, which are subject to a deformation parameter t. We derive systems of differential equations and give Lax pairs, yielding nonlinear differential equations in t for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a deformed Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlevé type P. The particular case of P arising here has the same four parameters as the solution found by Magnus (1995 J. Comput. Appl. Math. 57 215–37) but differs in the boundary conditions.

Linearization and connection coefficients of polynomial sequences: A matrix approach

For any sequence of polynomials {pk(t)} in one real or complex variable, where pk has degree k for k≥0, we find explicit expressions and recurrence relations for infinite matrices whose entries are the numbers d(n,m,k), called linearization coefficients, that satisfypn(t)pm(t)=∑k=0n+md(n,m,k)pk(t),n,m≥0. For any pair of polynomial sequences {uk(t)} and {pk(t)} we find infinite matrices whose entries are the numbers e(n,m,k) that satisfypn(t)pm(t)=∑k=0n+me(n,m,k)uk(t),n,m≥0. We also obtain recurrence relations and other properties of the linearization coefficients. Such results are obtained using only simple algebraic properties of infinite matrices. We apply the general results to general orthogonal polynomial sequences and to some simple families of orthogonal polynomials, such as the Chebyshev, Hermite, and Charlier families.

Characterization of orthogonal polynomials on lattices

We consider two sequences of orthogonal polynomials and such that with , and are sequences of complex numbers, , is the identity operator, x defines a lattice, and . We show that under some natural conditions, both involved orthogonal polynomials sequences and are semiclassical whenever k = m. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.

Quadratic Decomposition of Bivariate Orthogonal Polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Bäcklund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.

Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to RII-recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices.

Coherent pairs of moment functionals of the second kind and associated orthogonal polynomials and Sobolev orthogonal polynomials

Given a pair of quasi-definite moment functionals {v0,v1} we introduce the concept of coherence of the second kind in terms of an algebraic relation that the corresponding sequences of orthogonal polynomials satisfy. We characterize such moment functionals and give some illustrative examples taking into account they are semiclassical of class at most one. The relation between the corresponding monic Jacobi matrices is stated. For a pair of moment functionals satisfying the coherence property of the second kind, a Sobolev inner product is introduced. The connection formulas between the sequence of monic orthogonal polynomials associated with such a Sobolev inner product and the sequence of monic polynomials orthogonal with respect to the moment functional v0 are given.

Characterization of the D-Laguerre–Hahn orthogonal polynomials of class one via the cubic decomposition

In the present work, we deal with a monic orthogonal polynomial sequence (MOPS) deduced by its cubic decomposition (CD) . Our goal is to determine all D-Laguerre–Hahn strict of class one of these sequences. From the CD structure, only one family of MOPS appears in connection with the D-Laguerre–Hahn of class zero of Jacobi type in the singular case and the nonsingular one. Their recurrence coefficients are highlighted.