Multidimensional generalization of the Korous inequality

This paper deals with some qualitative properties of orthogonal polynomials in several variables. The boundedness and relations between two sets of orthonormal polynomials associated with an arbitrary weight function and its extension are investigated. It presents an analogy to Korous' result for general orthogonal polynomials in one variable.     

Recurrence relations for orthogonal rational functions

It is well known that members of families of polynomials, that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis, satisfy a three-term recursion relation. Analogous recursion formulas are available for orthogonal Laurent polynomials with a pole at the origin. This paper investigates recursion relations for orthogonal rational functions with arbitrary prescribed real or complex conjugate poles. The number of terms in the recursion relation is shown to be related to the structure of the orthogonal rational functions.     

A special class of polynomials orthogonal on the unit circle including the associated polynomials

Let (P _ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ω_υ) the monic associated polynomials of (P _v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (AP_υλ+BΩ_υλ)/A^λ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP _ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP _ν and of the corresponding orthogonality measure and Szegö function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.     

Biorthogonal Laurent Polynomials,Toplitz Determinants,Minimal Toda Orbits and Isomonodromic Tau Functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae. For all these classes of polynomials a 2 × 2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo-measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) Toplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.     

Linearization of the product of symmetric orthogonal polynomials

A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of₃ F ₂ and₃Φ₂ (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.     

Elliptic polynomials orthogonal on the unit circle with a dense point spectrum

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of the Jacobi elliptic functions. We find explicit expression for these polynomials in terms of elliptic hypergeometric functions. We show that the obtained polynomials are orthogonal on the unit circle with respect to a dense point measure. We also construct corresponding explicit systems of polynomials orthogonal on the interval of the real axis with respect to a dense point measure. They can be considered as an elliptic generalization of the Askey-Wilson polynomials of a special type.      

Formal orthogonal polynomials and Hankel/Toeplitz duality

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szeg recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.     

Rational solutions of the classical Boussinesq system

Rational solutions of the classical Boussinesq system are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation, which arises as a scaling reduction of the classical Boussinesq system. Generalized rational solutions of the classical Boussinesq system, which involve an infinite number of arbitrary constants, are also derived. The generalized rational solutions are analogues of such solutions for the Korteweg–de Vries, Boussinesq and nonlinear Schrödinger equations.     

Matrix Valued Orthogonal Polynomials Arising from the Complex Projective Space

The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer parameters to obtain an immediate extension beyond these cases.     

The asymptotics of interlacing sequences and the growth of continual young diagrams

We obtain a series of concrete results establishing a somewhat unexpected connection between the asymptotic representation theory of symmetric groups and the classical results for one-dimensional problems of mathematical physics and function theory. In particular:

1. The universal character of the division of roots for a wide class of orthogonal polynomials is shown.
2. A connection between the Plancherel measure of the infinite symmetric group and Markov's moment problem is established.
3. Asymptotics of the Plancherel measure turns out to be connected with the soliton-like solution of the simplest quasilinear equation, R′_t+RR′_x=0. Bibliography: 14 titles.

     

A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on (−∞,∞) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.     

Compound model of the generalized oscillator. I

We study the problem of realization of a given generalized oscillator by a system of N generalized oscillators of a different type. We consider a generalized oscillator related to a fixed system of orthogonal polynomials that are determined by three-term recurrent relations and the corresponding three-diagonal Jacobi matrix J. The case N =2 was considered in a previous paper. It was shown that in this case the orthogonality measure is symmetric with respect to rotation at angle π. In this paper, we consider the case N =3. We prove that such a problem has a solution only in two cases. In the first case, the Jacobi matrix related to the given “composite” generalized oscillator has block-diagonal form and consists of similar 3×3 blocks. In the second (more interesting) possible case, the Jacobi matrix is not block-diagonal. For this matrix, we construct the corresponding system of orthogonal polynomials. This system decomposes into three series which are related to Chebyshev polynomials of the second kind. The main result of the paper is a solution of the moment problem for the corresponding Jacobi matrix. In this case, the constructed measure is symmetric with respect to rotation at angle 2π/3. Bibliography: 6 titles.     

General multi-sum transformations and some implications

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence \(\{g(k)\}\)), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials and various multi-sums that are expressible as infinite products.     

Asymptotic behavior of the spectrum of combination scattering at Stokes phonons

For a class of polynomial quantum Hamiltonians used in models of combination scattering in quantum optics, we obtain the asymptotic behavior of the spectrum for large occupation numbers in the secondary quantization representation. Hamiltonians of this class can be diagonalized using a special system of polynomials determined by recurrence relations with coefficients depending on a parameter (occupation number). For this system of polynomials, we determine the asymptotic behavior a discrete measure with respect to which they are orthogonal. The obtained limit measures are interpreted as equilibrium measures in extremum problems for a logarithmic potential in an external field and with constraints on the measure. We illustrate the general case with an exactly solvable example where the Hamiltonian can be diagonalized by the canonical Bogoliubov transformation and the special orthogonal polynomials degenerate into the Krawtchouk classical discrete polynomials.     

Direct and inverse polynomial perturbations of hermitian linear functionals

This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree.The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm.Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.     

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.     

An algebraic interpretation of the multivariate <Emphasis Type="Italic">q</Emphasis>-Krawtchouk polynomials

The multivariate quantum q-Krawtchouk polynomials are shown to arise as matrix elements of “q-rotations” acting on the state vectors of many q-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations, and recurrence relations. The extension to an arbitrary number of variables is presented.     

Jucys–Murphy elements,orthogonal matrix integrals,and Jack measures

We study symmetric polynomials whose variables are odd-numbered Jucys–Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give their extension based on Jack polynomials.     

On the orthogonal polynomials in several variables

We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form.     

On Generating Discrete Orthogonal Bivariate Polynomials

In this paper we present an algorithm for recursively generating orthogonal bivariate polynomials on a discrete set S ². For this purpose we employ commuting pairs of real symmetric matrices H, K ^n×n to obtain, in a certain sense, a two dimensional Hermitian Lanczos method. The resulting algorithm relies on a recurrence having a slowly growing length. Practical implementation issues an applications are considered. The method can be generalized to compute orthogonal polynomials depending on an arbitrary number of variables.