An interpolation-theoretical characterization of the classical orthogonal polynomials

Acta Mathematica Hungarica -     

On classical orthogonal polynomials

Following the works of Nikiforov and Uvarov a review of the hypergeometric-type difference equation for a functiony(x(s)) on a nonuniform latticex(s) is given. It is shown that the difference-derivatives ofy(x(s)) also satisfy similar equations, if and only ifx(s) is a linear,q-linear, quadratic, or aq-quadratic lattice. This characterization is then used to give a definition of classical orthogonal polynomials, in the broad sense of Hahn, and consistent with the latest definition proposed by Andrews and Askey. The rest of the paper is concerned with the details of the solutions: orthogonality, boundary conditions, moments, integral representations, etc. A classification of classical orthogonal polynomials, discrete as well as continuous, on the basis of lattice type, is also presented.     

Point vortices and classical orthogonal polynomials

Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials.     

Multiple orthogonal polynomials for classical weights

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to 1$"> weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order . We also obtain explicit formulas and recurrence relations for these polynomials.

     

A new property of reproducing kernels for classical orthogonal polynomials

We exhibit a second-order differential operator commuting with the reproducing kernel $\text{∑}{}_{\mathrm{n\; ?\; 0}}{}^{\mathrm{T}}\text{φ}{}_{\mathrm{n}}\text{(λ)}\text{φ}{}_{\mathrm{n}}\text{(μ)}\text{h}{}_{\mathrm{n}}$ each time that {φ_n(λ)} is one of the classical orthogonal polynomials: Jacobi, Laguerre, Hermite and Bessel. This is the analog of a known property in the study of time and band-limited signals.     

The Dω—classical orthogonal polynomials

This is an expository paper; it aims to give an essentially self-contained overview of discrete classical polynomials from their characterizations by Hahn’s property and a Rodrigues’ formula which allows us to construct it. The integral representations of corresponding forms are given.     

Diophantine equations for classical continuous orthogonal polynomials

Let A, B, C denote rational numbers with AB ≠ 0 and m > n ≥ 3 arbitrary rational integers. We study the Diophantine equation AP_m(x) + Bp_n(y) = C, in x, y ? , where {Pk(x)}I is one of the three classical continuous orthogonal polynomial families, i.e. Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. We prove that with exception of the Chebyshev polynomials for all such polynomial families there are at most finitely many solutions (x, y) ? ² provided n > 4. The tools are besides the criterion [3], a theorem of Szeg— [14] on monotonicity of stationary points of polynomials which satisfy a second order Sturm-Liouville differential equation,

     

Characterizations of classical orthogonal polynomials on quadratic lattices

This paper is devoted to characterizations of classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal polynomials on quadratic lattices. Moreover a new matrix characterization of classical ortho-gonal polynomials in quadratic lattices is presented. From the Bochner type characterization we derive the three-term recurrence relation coefficients for these polynomials.     

Coefficients of multiplication formulas for classical orthogonal polynomials

In this paper using both analytic and algorithmic approaches, we derive the coefficients \(D_m(n,a)\) of the multiplication formula

$$\begin{aligned} p_n(ax)=\sum _{m=0}^nD_m(n,a)p_m(x) \end{aligned}$$

or the translation formula

$$\begin{aligned} p_n(x+a)=\sum _{m=0}^nD_m(n,a)p_m(x), \end{aligned}$$

where \(\{p_n\}_{n\ge 0}\) is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the \(q\)-classical orthogonal polynomials, as well as the classical orthogonal polynomials on a quadratic lattice and a \(q\)-quadratic lattice. We give a representation of the coefficients \(D_m(n,a)\) as a single, double or triple sum whereas in many cases we get simple representations.

     

Zeros of orthogonal Laurent polynomials and solutions of strong Stieltjes moment problems

The strong Stieltjes moment problem for a bisequence consists of finding positive measures μ with support in [0,∞) such that

     

Families of orthogonal Laurent polynomials,hyperelliptic lie algebras and elliptic integrals

We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties are listed in Section 3. We show that these families of polynomials in the variable t satisfy certain second-order linear differential equations that may be of interest to mathematicians in conformal field theory and number theory. We also prove that these families of polynomials in the setting of Date–Jimbo–Kashiwara–Miwa algebras when multiplied by a suitable power of t are orthogonal with respect to explicitly described kernels. Particular cases lead to new identities of elliptic integrals (see Section 5).     

Difference equations for discrete classical multiple orthogonal polynomials

For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a (r+1)th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal polynomials are given, which was already proved by Van Assche for the multiple Charlier polynomials and the multiple Meixner polynomials.     

On Koornwinder classical orthogonal polynomials in two variables

In 1975, Tom Koornwinder studied examples of two variable analogues of the Jacobi polynomials in two variables. Those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators. Some of these examples are well known classical orthogonal polynomials in two variables, such as orthogonal polynomials on the unit ball, on the simplex or the tensor product of Jacobi polynomials in one variable, but the remaining cases are not considered classical by other authors. The definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject. We analyze in detail Koornwinder polynomials and using the Koornwinder tools, new examples of orthogonal polynomials in two variables are given.     

Bounds for extreme zeros of some classical orthogonal polynomials

We use mixed three term recurrence relations typically satisfied by classical orthogonal polynomials from sequences corresponding to different parameters to derive upper (lower) bounds for the smallest (largest) zeros of Jacobi, Laguerre and Gegenbauer polynomials.