A set of orthogonal polynomials induced by a given orthogonal polynomial

Summary Given an integern 1, and the orthogonal polynomials _n (·; d) of degreen relative to some positive measured, the polynomial system induced by _n is the system of orthogonal polynomials corresponding to the modified measure . Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials from the recursion coefficients of the orthogonal polynomials belonging to the measured. A stable computational algorithm is proposed, which uses a sequence ofQR steps with shifts. For all four Chebyshev measuresd, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by Al-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros of _n and those of are studied for Gegenbauer measures, as well as the orthogonality—or lack thereof—of the polynomial sequence .Work supported in part by the National Science Foundation under grant DMS-9023403.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Classical orthogonal polynomials: A functional approach

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(u)=u. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.     

Matrix valued Jacobi polynomials

For every value of the parameters α,β>−1 we find a matrix valued weight whose orthogonal polynomials satisfy an explicit differential equation of Jacobi type.     

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L_n^(α,M,N) (x) orthogonal with respect to the Sobolev inner product

     

Fisher information of orthogonal polynomials I

Following the lead of J. Dehesa and his collaborators, we compute the Fisher information of the Meixner-Pollaczek, Meixner, Krawtchouk and Charlier polynomials.     

Some properties of generalized multiple Hermite polynomials

The purpose of this paper is to introduce and discuss a more general class of multiple Hermite polynomials. In this work, the explicit forms, operational formulas and a recurrence relation are obtained. Furthermore, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple Hermite polynomials.     

Discrete orthogonal polynomials and difference equations of several variables

The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.     

Zeros of linear combinations of Laguerre polynomials from different sequences

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely and . Proofs and numerical counterexamples are given in situations where the zeros of R_n, and S_n, respectively, interlace (or do not in general) with the zeros of , , k=n or n−1. The results we prove hold for continuous, as well as integral, shifts of the parameter α.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

Divisibility properties of generalized Laguerre polynomials

In this paper we give effective upper bounds for the degree k of divisors (over ?) of generalized Laguerre polynomials L^α_n(x), i.e. of for α = −tn − s − 1 and α = tn + s with t,s ∈ ?, t = O(log k), s = O(k log k) and k sufficiently large.     

Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications

Generalised matrix elements of the irreducible representations of the quantum SU(2) group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of sl(2). A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. Ser. A 66 (1990), 146–149] describing the generalised matrix elements in terms of the full four-parameter family of Askey-Wilson polynomials is given. Various known and new applications of this interpretation are presented.Supported by a NATO-Science Fellowship of the Netherlands Organization for Scientific Research (NWO).     

Convexity of the zeros of some orthogonal polynomials and related functions

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.     

A characterization of the classical orthogonal discrete and q-polynomials

In this paper we present a new characterization for the classical discrete and q$q$-classical (discrete) polynomials (in the Hahn's sense).     

Spreading lengths of Hermite polynomials

The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation is computationally proved. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments, mentioned previously, are posed.     

The asymptotics of the generalised Hermite-Bell polynomials

The Hermite-Bell polynomials are defined by for n=0,1,2,… and integer r≥2 and generalise the classical Hermite polynomials corresponding to r=2. We obtain an asymptotic expansion for as n→∞ using the method of steepest descents. For a certain value of x, two saddle points coalesce and a uniform approximation in terms of Airy functions is given to cover this situation. An asymptotic approximation for the largest positive zeros of is derived as n→∞. Numerical results are presented to illustrate the accuracy of the various expansions.     

A semiclassical perspective on multivariate orthogonal polynomials

Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi-orthogonality conditions. We obtain several characterizations for these polynomials including the analogues of the semiclassical Pearson differential equation, the structure relation and a differential-difference equation.