Abstract: 4 Where do orthogonal polynomials come from? 10 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Padé approximation and rational interpolation . . . . . . . . . . 11 Moment problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Jacobi matrices and spectral theory of self-adjoint operators . . . 13 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Abstract: Positive and sum-of-squares polynomials have received a special interest in the latest decade, due to their connections with semidefinite programming. Thus, efficient optimization methods can be employed to solve diverse problems involving polynomials. This book gathers the main recent results on positive trigonometric polynomials within a unitary framework; the theoretical results are obtained partly from the general theory of real polynomials, partly from self-sustained developments. The optimization applications cover a field different from that of real polynomials, mainly in signal processing problems: design of 1-D and 2-D FIR or IIR filters, design of orthogonal filterbanks and wavelets, stability of multidimensional discrete-time systems. Positive Trigonometric Polynomials and Signal Processing Applicationshas two parts: theory and applications. The theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The presentation starts by giving the main results for univariate polynomials, which are later extended and generalized for multivariate polynomials. The applications part is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semidefinite programming form, ready to be solved with algorithms freely available, like those from the library SeDuMi.

Abstract: Introduction to $q$-analogues and symmetric functions Macdonald polynomials and the space of diagonal harmonics The $q, t$-Catalan numbers The $q, t$-Schroder polynomial Parking functions and the Hilbert series The shuffle conjecture The proof of the $q, t$-Schroder theorem The combinatorics and Macdonald polynomials The Loehr-Warrington conjecture Solutions to exercises Bibliography.

Abstract: This paper shows how Hahn moments provide a unified understanding of the recently introduced Chebyshev and Krawtchouk moments. The two latter moments can be obtained as particular cases of Hahn moments with the appropriate parameter settings and this fact implies that Hahn moments encompass all their properties. The aim of this paper is twofold: (1) To show how Hahn moments, as a generalization of Chebyshev and Krawtchouk moments, can be used for global and local feature extraction and (2) to show how Hahn moments can be incorporated into the framework of normalized convolution to analyze local structures of irregularly sampled signals.

Abstract: The real Ginibre ensemble consists of random N x N matrices formed from independent and identically distributed standard Gaussian entries. By using the method of skew orthogonal polynomials, the general n-point correlations for the real eigenvalues, and for the complex eigenvalues, are given as n x n Pfaffians with explicit entries. A computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented. This is relevant to May's stability analysis of biological webs.

Abstract: Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

Abstract: We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e−V(x) where V is a polynomial, V(x) = κ2mx2m + · · ·, κ2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [12], and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel-Rotach-type asymptotics for the orthogonal polynomials found in [12] plays a central role. The formulae in [46] express the correlation kernels for β = 1, 4 as a sum of a Christoffel-Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [9], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [49]. © 2006 Wiley Periodicals, Inc.

Abstract: This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl--Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrodinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete--Szegő theory.

Abstract: We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally C 2 and locally C 3 function (see Theorem 3.1). The proof as our previous proof in (Pastur and Shcherbina in J. Stat. Phys. 86:109–147, 1997) is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the sin -kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper (Boutet de Monvel, et al. in J. Stat. Phys. 79:585–611, 1995) on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.

Abstract: In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.

Abstract: Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also study the relationship between Stieltjes-Wigert and Rogers-Szego polynomials and the corresponding equivalence with a unitary matrix model. Finally, we give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

Abstract: We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.

Abstract: A nonpolynomial one-dimensional quantum potential representing an oscillator, that can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends of a parameter $a$, is considered and then a particular case is studied with great detail. It is proven that it is Schr\"odinger solvable and then the wave functions $\Psi_n$ and the energies $E_n$ of the bound states are explicitly obtained. Finally it is proven that the solutions determine a family of orthogonal polynomials ${\cal P}_n(x)$ related with the Hermite polynomials and such that: (i) Every ${\cal P}_n$ is a linear combination of three Hermite polynomials, and (ii) They are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.

Abstract: In the recent paper the interesting q-Euler numbers and polynomials introduced in JMAA. The purpose of this paper is to construct the modified q-Euler numbers and polynomiasl. Finally we will give the interesting many identities related to these numbers and polynomials.

Abstract: We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t eq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

Abstract: We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.

Abstract: A nonlinear model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature is exactly solved. This model depends of a parameter $\la$ that is related with the curvature of the space. Firstly the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a $\la$-dependent invariant measure $d\mu_\la$ and expressing the Hamiltonian as a function of the Noether momenta. In the second part the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrodinger equation is studied. Two $\la$-dependent Sturm-Liouville problems, related with two different $\la$-deformations of the Hermite equation, are obtained. This leads to the study of two $\la$-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions $\Psi_{m,n}$ and the energies $E_{m,n}$ of the bound states are exactly obtained in both the sphere $S^2$ and the hyperbolic plane $H^2$.

Abstract: We extend the results of Denisov-Rakhmanov, Szego-Shohat-Nevai, and Killip-Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.

Abstract: The Padua points are a family of points on the square [−1, 1]2 given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The Lp convergence of the interpolation polynomials is also studied.

Abstract: A general theoretical approach has been developed for the determination of orthonormal polynomials over any integrable domain, such as a hexagon. This approach is better than the classical Gram-Schmidt orthogonalization process because it is nonrecursvie and can be performed rapidly with matrix transformations. The determination of the orthonormal hexagonal polynomials is demonstrated as an example of the matrix approach.