Families Of Orthogonal Polynomials Research Articles

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.

Monomial and Rodrigues orthogonal polynomials on the cone

We study two families of orthogonal polynomials with respect to the weight function w(t)(t2−‖x‖2)μ−12, μ>−12, on the cone {(x,t):‖x‖≤t,x∈Rd,t>0} in Rd+1. The first family consists of monomial polynomials Vk,n(x,t)=tn−|k|xk+⋯ for k∈N0d with |k|≤n, which has the least L2 norm among all polynomials of the form tn−|k|xk+P with deg⁡P≤n−1, and we will provide an explicit construction for Vk,n. The second family consists of orthogonal polynomials defined by the Rodrigues type formulas when w is either the Laguerre weight or the Jacobi weight, which satisfies a generating function in both cases. The two families of polynomials are partially biorthogonal.

On a Karhunen-Loève expansion based on Krawtchouk polynomials with application to Bahadur optimality for the binomial location family

In this paper we extend to the discrete case a Karhunen-Lo?ve expansion already known for continuous families of classical orthogonal polynomials. This expansion involves Krawtchouk polynomials. It provides us with the orthogonal decomposition of the covariance function of a weighted discrete Brownian bridge process. We introduce a discrete Cram?r-von Mises statistic associated with this covariance function. We show that this statistic satisfies a property of Bahadur local optimality for a statistical test in the location family for binomial distributions. Our statistic and the goodness-of-fit problem we deal with can be seen as a discrete version of a problem stated by Y. Nikitin about the statistic of de Wet and Venter. Our proofs make use of the formulas valid for all classical orthogonal families of polynomials, so that the way most of our results can be extended to Meixner, Hahn, and Charlier polynomials and the associated distributions is clearly outlined.

On polynomial transformations preserving purely imaginary zeros*

In this present work polynomial transformations are identified that preserve the property of the polynomials having all zeros lying on the imaginary axis. Existence results concerning families of polynomials whose generalized Mellin transforms have zeros all lying on the critical line are then derived. Inherent structures are identified from which a simple proof relating to the Gegenbauer family of orthogonal polynomials is subsequently deduced. Some discussion about the choice of generalized Mellin transform is also given.

Accurate computations with totally positive matrices applied to the computation of Gaussian quadrature formulae

For some families of classical orthogonal polynomials defined on appropriate intervals, it is shown that the corresponding Jacobi matrices are totally positive and their bidiagonal factorizations can be accurately computed. By exploiting these facts, an algorithm to compute with high relative accuracy the eigenvalues of those Jacobi matrices, and consequently the nodes of Gaussian quadrature formulae for those families of orthogonal polynomials, is presented. An algorithm is also presented for the computation of the eigenvectors of these Jacobi matrices, and hence the weights of Gaussian quadrature formulae. Although in this case high relative accuracy is not theoretically guaranteed, the numerical experiments with our algorithm provide very accurate results.

Refined multilayered beam, plate and shell elements based on Jacobi polynomials

In this paper, theories of structures based on hierarchical Jacobi expansions are explored for the static analysis of multilayered beams, plates and shells. They belong to the family of classical orthogonal polynomials. This expansion is employed in the framework of the Carrera Unified Formulation (CUF), which allows to generate finite element stiffness matrices in a straightforward way. CUF allows also to employ both layer-wise and equivalent single layer approaches in order to obtain the desired degree of precision and computational cost. In this work, CUF is exploited for the analysis of one-dimensional beams and two-dimensional plates and shells, and several case studies from the literature are analysed. Displacements, in-plane, transverse and shear stresses are shown. In particular, for some benchmarks, the shear stresses are calculated using the constitutive relations and the stress recovery technique. The obtained results clearly show the convenience of using equivalent single layer models when calculating displacements, in-plane stresses and shear stresses recovered by three-dimensional indefinite equilibrium equations. On the other hand, layer-wise models are able to accurately predict the structural behaviour, even though higher degrees of freedom are needed.

Approximation and Localized Polynomial Frame on Double Hyperbolic and Conic Domains

We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in Xu (J Funct Anal 281(12):109257, 2021) for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed-form formulas for the reproducing kernels. The main results provide construction of semi-discrete localized tight frame in weighted $$L^2$$ norm and characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz–Zygmund inequalities, positive cubature rules, Christoffel functions, and Bernstein inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.

Multiple orthogonal polynomials: Pearson equations and Christoffel formulas

Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss–Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre–Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi–Piñeiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi–Piñeiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes–Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial.

Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials.

The univariate noncentral distributions can be derived by multiplying their central distributions with translation factors. When constructed in terms of translated uniform distributions on unit radius hyperspheres, these translation factors become generating functions for classical families of orthogonal polynomials. The ultraspherical noncentral t, normal N, F, and distributions are thus found to be associated with the Gegenbauer, Hermite, Jacobi, and Laguerre polynomial families, respectively, with the corresponding central distributions standing for the polynomial family-defining weights. Obtained through an unconstrained minimization of the Gibbs potential, Jaynes’ maximal entropy priors are formally expressed in terms of the empirical densities’ entropic convex duals. Expanding these duals on orthogonal polynomial bases allows for the expedient determination of the Jaynes–Gibbs priors. Invoking the moment problem and the duality principle, modelization can be reduced to the direct determination of the prior moments in parametric space in terms of the Bayes factor’s orthogonal polynomial expansion coefficients in random variable space. Genomics and geophysics examples are provided.

Monic Bivariate Polynomials on Quadratic and q-Quadratic Lattices

Monic families of bivariate Askey–Wilson polynomials and q-Racah polynomials are explicitly given. Monic families of bivariate orthogonal polynomials, both in quadratic and q-quadratic lattices, are also explicitly given using appropriate limit relations.

A family of orthogonal polynomials corresponding to Jacobi matrices with a trace class inverse

Assume that {an;n≥0} is a sequence of positive numbers and ∑an−1<∞. Let αn=kan, βn=an+k2an−1 where k∈(0,1) is a parameter, and let {Pn(x)} be an orthonormal polynomial sequence defined by the three-term recurrence α0P1(x)+(β0−x)P0(x)=0,αnPn+1(x)+(βn−x)Pn(x)+αn−1Pn−1(x)=0for n≥1, with P0(x)=1. Let J be the corresponding Jacobi (tridiagonal) matrix, i.e. Jn,n=βn, Jn,n+1=Jn+1,n=αn for n≥0. Then J−1 exists and belongs to the trace class. We derive an explicit formula for Pn(x) as well as for the characteristic function of J and describe the orthogonality measure for the polynomial sequence. As a particular case, the modified q-Laguerre polynomials are introduced and studied.

Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.

On the families of polynomials forming a part of the Askey–Wilson scheme and their probabilistic applications

In this paper, we review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey–Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial rôle in the attempts to generalize Gaussian distributions and find their bounded support generalizations. This paper is based on 65 items in the predominantly recent literature.

Classifying families of orthogonal polynomials having Galois group the alternating group

We give a classification of the generalized Laguerre polynomials Ln(α)(x), where α≠n is an integer, having the alternating group An as the Galois group over the rationals for up to a finitely many exceptions.

Algebraic and complexity‐like properties of Jacobi polynomials: Degree and parameter asymptotics

AbstractThe Jacobi polynomials conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two‐parameter weight function on the interval . The spreading of its associated probability density (i.e., the Rakhmanov density) over the support interval has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic (Shannon and Rényi entropies) and the monotonic complexity‐like measures of Cramér–Rao, Fisher–Shannon, and LMC (López‐Ruiz, Mancini, and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non‐handy way; specially when the degree or the parameters are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity‐like properties of the Jacobi polynomials in the two extreme situations: (; fixed ) and (; fixed ). These two asymptotics are relevant per se and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly, and quasi‐exactly solvable quantum‐ mechanical potentials which model numerous atomic and molecular systems.

Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to 2times 2 Hypergeometric Operators

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size 2times 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.