Orthogonal polynomials and some q-beta integrals of Ramanujan

Two integrals of Ramanujan are used to define a q-analogue of the Euler beta integral on the real line and of the Cauchy beta-integral on the complex unit circle. Such integrals are connected to orthogonal, biorthogonal and Laurent polynomials. Explicit examples of Laurent orthogonal polynomials are given on the real line and on the circle.     

Fredholm Determinant Evaluations of the Ising Model Diagonal Correlations and their λ Generalization

The diagonal spin–spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants—one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural λ‐parameter generalization and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case [ 1 ], applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi‐orthogonal situation.     

CMV Matrices and Little and Big ?1 Jacobi Polynomials

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related to the theory of CMV matrices. It contains an arbitrary parameter ?? which leads to a linear operator pencil. We show that the little and big ?1?Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.     

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.     

Properties of interpolatory quadrature with equidistant nodes on the unit circle

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.     

On the continuous analog of Rakhmanov's Theorem for orthogonal polynomials

We obtain the continuous analogs of Rakhmanov's Theorem for polynomials orthogonal on the unit circle. Sturm-Liouville operators and Krein systems are considered. For a Sturm-Liouville operator with bounded potential q, we prove the following statement. If the essential spectrum and absolutely continuous component of the spectral measure fill the whole positive half-line, then q decays at infinity in the certain integral sense.     

Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle

Rakhmanov's theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle.     

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.     

On extensions of a theorem of Baxter

We combine the Riemann–Hilbert approach with the techniques of Banach algebras to obtain an extension of Baxter's Theorem for polynomials orthogonal on the unit circle. This is accomplished by using the link between the negative Fourier coefficients of the scattering function and the coefficients in the recurrence formula satisfied by these polynomials.     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

A Cubic Decomposition of Sequences of Orthogonal Polynomials on the Unit Circle

In this paper we will discuss the problem of generation of sequences of orthogonal polynomials with respect to measures supported on the unit circle from a given sequence of orthogonal polynomials using a perturbation of a cubic sieved process. The basic tools are the Szeg? forward recurrence relation as well as the fact of the coprimality of orthogonal polynomials on the unit circle and their corresponding reverse polynomials. We also give the connection between the associated orthogonality measures. Finally, some examples of this cubic decomposition are shown.     

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.      

Orthogonal polynomials on the unit circle and their derivatives

We characterize the sequences of orthogonal polynomials on the unit circle whose derivatives are also orthogonal polynomials on the unit circle. Some relations for the sequences of derivatives of orthogonal polynomials are provided. Finally, we pose some problems about orthogonality-preserving maps and differential equations for orthogonal polynomials on the unit circle.     

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.     

Asymptotics for polynomials orthogonal over the unit disk with respect to a positive polynomial weight

We derive asymptotics for polynomials orthogonal over the complex unit disk with respect to a weight of the form ²|h(z)|, with h(z) a polynomial without zeros in |z|<1. The behavior of the polynomials is established at every point of the complex plane. The proofs are based on adapting to the unit disk a technique of J. Szabados for the asymptotic analysis of polynomials orthogonal over the unit circle with respect to the same type of weight.     

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.     

Quadrature formulas on the unit circle based on rational functions

Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.     

Multiple Orthogonal Polynomials on the Unit Circle

We introduce multiple orthogonal polynomials on the unit circle. We show how this is related to simultaneous rational approximation to Caratheodory functions (two-point Hermite-Pade approximation near zero and near infinity). We give a Riemann-Hilbert problem for which the solution is in terms of type I and type II multiple orthogonal polynomials on the unit circle, and recurrence relations are obtained from this Riemann-Hilbert problem. Some examples are given to give an idea of the behavior of the zeros of type II multiple orthogonal polynomials.     

Root statistics of random polynomials with bounded Mahler measure

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yield a Pfaffian point process on the complex plane. When N is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e.   scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of ±1 which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of C$\mathbb{C}$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.