Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials (original) (raw)

Review of Some Askey-Scheme of Hypergeometric orthogonal Polynomials

The present paper is a review of different kind of Orthogonal functions with there important properties, main emphasize is given on the hypergeometric function. In this paper we have discussed orthogonality properties of orthogonal polynomials like Rodrigues formula, Recurrence relation, Generating function classification of Orthogonal polynomials using lemmas and examples. This paper also deals with all these properties of Laguerre, Hermite polynomials.

Quasi-orthogonality of some hypergeometric polynomials

Integral Transforms and Special Functions

In this paper, we prove the quasi-orthogonality of a family of 2 F 2 polynomials and several classes of 3 F 2 polynomials that do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, as a special case, two 3 F 2 polynomials considered by Dickinson in 1961. We also discuss the location and interlacing of the real zeros of our polynomials.

Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

Journal of Computational and Applied Mathematics, 2010

The asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a Lauricella function. The polynomials are defined by means of a three-term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices. Subclasses of systems of orthogonal polynomials having their contracted measure of zeros of regular, uniform, Wigner, Weyl, Karamata and hypergeometric types are explicitly identified. Some illustrative examples are given.

The Role of Hermite Polynomials in Asymptotic Analysis

Special Functions - Proceedings of the International Workshop, 2000

Hermite polynomials are considered as appraximants in asymptotic representations of certain other polynomials. Examples are given for polynomials from the Askey scheme of hypergeometric orthogonal polynomials. We also mention that Hermite polynomials can be used as main approximants in uniform asymptotic representations of certain types of integrals and differential equations.

On the Askey-Wilson polynomials

Constructive Approximation, 1992

Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach. In [3], [4], and [16] a number of new systems of orthogonal polynomials (i.e., 4F3 and 4q~3 polynomials) were introduced. The main properties of these polynomials have been established on the basis of the theory of hypergeometric and basic hypergeometric series [9], [11]. As is clear now, the Askey-Wilson [3], [4] and Wilson [16] polynomials are the most general extensions of the Jacobi, Laguerre, and Hermite polynomials known at present. The families of polynomials, considered in [1], [3], [4], and [16], together with their special and limiting cases, form a mathematical entity-the classical orthogonal polynomials of a discrete variable on nonuniform lattices [1], [15]. The introduction of such an object made it possible to generalize all the fundamental properties characteristic of Hahn's approach [12], [10]. We shall disc/ass these properties for the most general system-the Askey-Wilson polynomials, in the framework of an approach developed in [15], [7]. 1. Hypergeometrie-Type Difference Equation Consider the hypergeometric-type difference equation in the self-adjoint form (for details, see [15]) A [cr(z)p(z)~l+2p(z)y(z)=O ' (1) Vxl(z) (2) A[cr(z)p(z)] = p(z)~(z)Vx~(z), on the lattice x(z) = cosh 2o9z = ~q" + q-Z), q = e2~,, where xl(z) = x(z + 89 and AT(z) = Vf(z + 1) = f(z + 1)-f(z).

Complexity analysis of hypergeometric orthogonal polynomials

Journal of Computational and Applied Mathematics, 2015

The complexity measures of the Crámer-Rao, Fisher-Shannon and LMC (López-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density ρ n (x) = ω(x)p 2 n (x) of the polynomials p n (x) orthogonal with respect to the weight function ω(x), x ∈ (a, b), are used to quantify various twofold facets of the spreading of the Hermite, Laguerre and Jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways. Their explicit (Crámer-Rao) and asymptotical (Fisher-Shannon, LMC) values are given for the three systems of orthogonal polynomials. Then, these complexity-type mathematical quantities are numerically examined in terms of the polynomial's degree n and the parameters which characterize the weight function. Finally, several open problems about the generalised hypergeometric functions of Lauricella and Srivastava-Daoust types, as well as on the asymptotics of weighted L q-norms of Laguerre and Jacobi polynomials are pointed out.

Orthogonal Polynomials of Askey-Wilson Type

arXiv (Cornell University), 2022

We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on R. A limiting case of this family is an infinite system of orthogonal polynomials whose moment problem is indeterminate. We provide several orthogonality measures for the infinite family and derive their Plancherel-Rotach asymptotics.

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Symmetry, Integrability and Geometry: Methods and Applications, 2018

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its q-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big q-Jacobi polynomials and big q-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.