Families Of Orthogonal Polynomials Research Articles

Unique special solution for discrete Painlevé II

We show that the discrete Painlevé II equation with starting value a − 1 = − 1 has a unique solution for which − 1 < a n < 1 for every n ≥   0 . This solution corresponds to the Verblunsky coefficients of a family of orthogonal polynomials on the unit circle. This result was already proved for certain values of the parameter in the equation and recently a full proof was given by Duits and Holcomb [A double scaling limit for the d-PII equation with boundary conditions. arXiv:2304.02918 [math.CA]]. In the present paper we give a different proof that is based on an idea put forward by Tomas Lasic Latimer [Unique positive solutions to q-discrete equations associated with orthogonal polynomials, J. Difference Equ. Appl. 27 (2021), pp. 763–775.] which uses orthogonal polynomials. We also give an upper bound for this special solution.

Nearest Neighbor Recurrence Relations for Meixner–Angelesco Multiple Orthogonal Polynomials of the Second Kind

This paper studies a new family of Angelesco multiple orthogonal polynomials with shared orthogonality conditions with respect to a system of weight functions, which are complex analogs of Pascal distributions on a legged star-like set. The emphasis is placed on the algebraic properties, such as the raising operators, the Rodrigues-type formula, the explicit expression of the polynomials, and the nearest neighbor recurrence relations.

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

A bivariate Q-polynomial structure for the non-binary Johnson scheme

The notion of multivariate P- and Q-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate P-polynomial association scheme. We show here that it is also a bivariate Q-polynomial association scheme for some parameters. This provides, with the P-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.

Time and band limiting for exceptional polynomials

The “time-and-band limiting” commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones.Here we give a general result that insures the existence of a commuting differential operator for a given family of exceptional orthogonal polynomials satisfying the “bispectral property”. As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials. We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.

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.

On the extended weight monoid of a spherical homogeneous space and its applications to spherical functions

Given a connected simply connected semisimple group G G and a connected spherical subgroup K ⊆ G K\subseteq G we determine the generators of the extended weight monoid of G / K G/K , based on the homogeneous spherical datum of G / K G/K . Let H ⊆ G H\subseteq G be a reductive subgroup and let P ⊆ H P\subseteq H be a parabolic subgroup for which G / P G/P is spherical. A triple ( G , H , P ) (G,H,P) with this property is called multiplicity free system and we determine the generators of the extended weight monoid of G / P G/P explicitly in the cases where ( G , H ) (G,H) is strictly indecomposable. The extended weight monoid of G / P G/P describes the induction from H H to G G of an irreducible H H -representation π : H → GL ⁡ ( V ) \pi :H\to \operatorname {GL}(V) whose lowest weight is a character of P P . The space of regular End ⁡ ( V ) \operatorname {End}(V) -valued functions on G G that satisfy F ( h 1 g h 2 ) = π ( h 1 ) F ( g ) π ( h 2 ) F(h_{1}gh_{2})=\pi (h_{1})F(g)\pi (h_{2}) for all h 1 , h 2 ∈ H h_{1},h_{2}\in H and all g ∈ G g\in G is a module over the algebra of H H -biinvariant regular functions on G G . We show that under a mild assumption this module is freely and finitely generated. As a consequence the spherical functions of such a type π \pi can be described as a family of vector-valued orthogonal polynomials with properties similar to Jacobi polynomials.

DEK-type orthogonal polynomials and a modification of the Christoffel formula

In this note we revisit one of the first known examples of exceptional orthogonal polynomials that was introduced by Dubov, Eleonskii, and Kulagin in relation to nonharmonic oscillators with equidistant spectra. We dissect the DEK polynomials using the discrete Darboux transformations and unravel a characterization bypassing the differential equation that defines the DEK polynomials. This characterization also leads to a family of general orthogonal polynomials with missing degrees and this approach manifests its relation to biorthogonal polynomials introduced by Iserles and Nørsett, which are applicable to a whole range of problems in computational and applied analysis. We also obtain a modification of the Christoffel formula for this family since its classical form cannot be applied in this case.

Norm-Attainable Polynomials: Characterizations and Properties in Orthogonal Polynomial Families

Norm-attainable polynomials play a crucial role in various mathematical contexts. This research paper investigates the characterizations and properties of norm-attainable polynomials in different orthogonal polynomial families. We explore their behavior, convexity, and positive definiteness. Additionally, we establish their norm-attainability in specific intervals for various weight functions. The findings contribute to a deeper understanding of norm-attainable polynomials and their applications in approximation theory and mathematical modeling.

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.

Multiple orthogonal polynomials associated with the exponential integral

AbstractWe introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights on the positive real line, with the gamma density and a density related to the exponential integral . We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed‐type multiple orthogonal polynomials.

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.

Classical Orthogonal Polynomials Revisited

This manuscript contains a small portion of the algebraic theory of orthogonal polynomials developed by Maroni and their applicability to the study and characterization of the classical families, namely Hermite, Laguerre, Jacobi, and Bessel polynomials. It is presented a cyclical proof of some of the most relevant characterizations, particularly those due to Al-Salam and Chihara, Bochner, Hahn, Maroni, and McCarthy. Two apparently new characterizations are also added. Moreover, it is proved through an equivalence relation that, up to constant factors and affine changes of variables, the four families of polynomials named above are the only families of classical orthogonal polynomials.

Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator Δ. Concretely, we treat the generalized Charlier weights in the framework of Δ-Sobolev orthogonality. We obtain an asymptotic expansion for these orthogonal polynomials where the falling factorial polynomials play an important role.

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.

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.

Explicit Relation Between Two Resolvent Matrices of the Truncated Hausdorff Matrix Moment Problem

We consider the explicit relation between two resolvent matrices related to the truncated Hausdorff matrix moment problem (THMM) in the case of an even and odd number of moments. This relation is described with the help of four families of orthogonal matrix polynomials on the finite interval [a, b] and their associated second kind polynomials.

Extracting edge stress intensity function for isotropic 3D cracked domains having stochastic material properties

The Stress Intensity Factors (SIFs) are fundamental and valuable parameters in fracture mechanics since many failure criteria involve them. In the case of a 3D crack, the SIFs extraction, based on a post-processing solution using the Finite-Element Method (FEM), is performed repeatably at several points along the singular edge (as a point-wise method) to evaluate the functional representation of the SIF along the crack edge. The Quasi Dual Function Method (QDFM) provides a polynomial approximation of the SIFs along the edge and eliminates the need for point-wise repeated calculations along the edge. Herein, we extend the QDFM to provide a broader 3D extraction method that includes stochasticity in the material properties in the case of an isotropic cracked domain having a straight edge, using the generalized Polynomial chaos (gPC). We apply the 2D gPC SIF extraction method over the QDFM to provide a stochastic Edge Stress Intensity Function (ESIF) polynomial approximation for 3D cracked domains with stochastic material properties (stochastic variables). The obtained ESIF is expressed using three families of orthogonal polynomials: Jacobi polynomials which approximate the deterministic solution of the ESIF (as part of the QDFM) and two families of orthogonal polynomials, selected by the probability distribution function of the material properties. The proposed method provides a functional polynomial presentation of the stochastic ESIF along the crack edge. Numerical example problems are provided where the stochastic approximation of the ESIF is computed. The stochastic properties of the ESIF: Mean, Variance, Skewness and Kurtosis are calculated, and the convergence of these properties is examined. The stochastic approximation of the ESIF is compared with results obtained using the 2D point-wise gPC SIFs extraction method. The proposed method is very efficient compared with the known point-wise methods since it provides a continuous function along the singular edge, unlike the point methods that require a new calculation for each point separately. The numerical examples demonstrate the robustness and high accuracy of the proposed method.

Two-body Coulomb problem and g(2) algebra (once again about the Hydrogen atom)

Taking the Hydrogen atom as an example it is shown that if the symmetry of a three-dimensional system is O(2)⊕Z2, the variables (r,ρ,φ) allow a separation of the variable φ, and the eigenfunctions define a new family of orthogonal polynomials in two variables, (r,ρ2). These polynomials are related to the finite-dimensional representations of the algebra gl(2)⋉R3∈g(2) (discovered by S Lie around 1880 which went almost unnoticed), which occurs as the hidden algebra of the G2 rational integrable system of 3 bodies on the line with 2- and 3-body interactions (the Wolfes model). Namely, those polynomials occur intrinsically in the study of the Zeeman effect on Hydrogen atom. It is shown that in the variables (r,ρ,φ) in the quasi-exactly-solvable generalized Coulomb problem new polynomial eigenfunctions in (r,ρ2)-variables are found.