Abstract: We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schr\"odinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.

Abstract: Recently introduced vertex-weighted Wiener polynomials are a generalization of both vertex-weighted Wiener numbers and ordinary Wiener polynomials. We present here explicit formulae for vertex-weighted Wiener polynomials of the most frequently encountered classes of composite graphs.

Abstract: A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.

Abstract: Inner Product Space- The Sturm-Liouville Theory- Fourier Series- Orthogonal Polynomials- Bessel Functions- The Fourier Transformation- The Laplace Transformation

Abstract: We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.

Abstract: We discuss various aspects of multi-instanton configurations in generic multi-cut matrix models. Explicit formulae are presented in the two-cut case and, in particular, we obtain general formulae for multi-instanton amplitudes in the one-cut matrix model case as a degeneration of the two-cut case. These formulae show that the instanton gas is ultra-dilute, due to the repulsion among the matrix model eigenvalues. We exemplify and test our general results in the cubic matrix model, where multi-instanton amplitudes can be also computed with orthogonal polynomials. As an application, we derive general expressions for multi-instanton contributions in two-dimensional quantum gravity, verifying them by computing the instanton corrections to the string equation. The resulting amplitudes can be interpreted as regularized partition functions for multiple ZZ-branes, which take into full account their back-reaction on the target geometry. Finally, we also derive structural properties of the trans-series solution to the Painleve I equation.

Abstract: In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erd˝ os and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.

Abstract: A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrodinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalization measures and the normalisation constants, etc., are determined explicitly.

Abstract: The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly $k$ eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly non-symmetric.

Abstract: The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler-Moser polynomials, which arise in the description of rational solutions of the Korteweg-de Vries equation. For quadupole background flow, vortex configurations are given by the zeros of polynomials expressed as wronskians of Hermite polynomials. Further new solutions are found in this case using the special polynomials arising the in the description of rational solutions of the fourth Painleve equation.

Abstract: Physical and mathematical applications of fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we have discovered and developed the fractional generalization of Bell polynomials, Bell numbers, and Stirling numbers. Appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been applied to evaluate skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been obtained. A representation of Schlafli polynomials in terms of fractional Stirling numbers of the second kind has been found. A new representations of Mittag-Leffler function involving fractional Bell polynomials and fractional Stirling numbers of the second kind have been discovered. Fractional Stirling numbers of the first kind have been introduced and studied. Two new polynomial sequences associated with fractional Poisson probability distribution have been launched and explored. The relationship between new polynomials and the orthogonal Charlier polynomials has also been investigated. In the limit case when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed developments and implementations turn into the well-known results of quantum optics, the theory of combinatorial numbers and the theory of orthogonal polynomials of discrete variable.

Abstract: Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equa

Abstract: We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes.

Abstract: The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly non-symmetric.

Abstract: The main object of this paper is to investigate the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials. We first establish two relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. It can be found that many results obtained before are special cases of these two relationships. Moreover, we have a study on the sums of products of the Apostol-Bernoulli polynomials and of the Apostol-Euler polynomials.

Abstract: Preface Index of authors List of participants Conference photograph, with key The trace problem for totally positive algebraic integers Julian Aguirre and Juan Carlos Peral, with an appendix by Jean-Pierre Serre Mahler's measure: from Number Theory to Geometry Marie Jose Bertin Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps Frits Beukers and Hans Montanus The merit factor problem Peter Borwein, Ron Ferguson and Joshua Knauer Barker sequences and flat polynomials Peter Borwein and Michael Mossinghoff The Hansen-Mullen primitivity conjecture: completion of proof Stephen Cohen and Mateja Presern An inequality for the multiplicity of the roots of a polynomial Arturas Dubickas Newman's inequality for increasing exponential sums Tamas Erdelyi On primitive divisors of n2 + b Graham Everest and Glyn Harman Irreducibility and greatest common divisor algorithms for sparse polynomials Michael Filaseta, Andrew Granville and Andrzej Schinzel Consequences of the continuity of the monic integer transfinite diameter Jan Hilmar Nonlinear recurrence sequences and Laurent polynomials Andrew Hone Conjugate algebraic numbers on conics: a survey James McKee On polynomial ergodic averages and square functions Radhakrishnan Nair Polynomial inequalities, Mahler's measure, and multipliers Igor E. Pritsker Integer transfinite diameter and computation of polynomials Georges Rhin and Qiang Wu Smooth divisors of polynomials Eira Scourfield Self-inversive polynomials with all zeros on the unit circle Christopher Sinclair and Jeffrey Vaaler The Mahler measure of algebraic numbers: a survey Chris Smyth.

Abstract: We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that the polynomials P_n(z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A_n, generated by d algebraically independent difference operators acting on the discrete variables n_1, n_2,..., n_d, which is also diagonalized by P_n(z). This leads to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.

Abstract: This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam-Chihara polynomials and a relation between these polynomials for different values of parameters.

Abstract: We deal with the problem of obtaining closed formulas for the connection coefficients between orthogonal polynomials and the canonical sequence. We use a recurrence relation fulfilled by these coefficients and symbolic computation with the Mathematica language. We treat the cases of Gegenbauer, Jacobi and a new semi-classical sequence.

Abstract: We point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads $(N_1,N_2)$ are analytically known from the Jacobi ensemble of Random Matrix Theory. Using this result and a simple linear statistic, we give an exact and non-perturbative expression for moments of the form $ $ for $m>-|N_1-N_2|-1$ and $\beta=2$, thus improving the existing results in the literature. Secondly, we offer an independent derivation of the average density and higher order correlation functions for $\beta=2,4$ which does not make use of the orthogonal polynomials technique. This result may be relevant for an efficient numerical implementation avoiding determinants.

Abstract: Least Square Methods The Least Square Algorithm Linear Least Square Methods Nonlinear Least Squares Algorithm Properties of Least Square Algorithms Examples Polynomial Approximation Gram-Schmidt Procedure of Orthogonalization Hypergeometric Function Approach to Generate Orthogonal Polynomials Discrete Variable Orthogonal Polynomials Approximation Properties of Orthogonal Polynomials Artificial Neural Networks for Input-Output Approximation Introduction Direction-Dependent Approach Directed Connectivity Graph Modified Minimal Resource Allocating Algorithm (MMRAN) Numerical Simulation Examples Multi-Resolution Approximation Methods Wavelets Bezier Spline Moving Least Squares Method Adaptive Multi-Resolution Algorithm Numerical Results Global-Local Orthogonal Polynomial MAPping (GLO-MAP) in N Dimensions Basic Ideas Approximation in 1, 2, and N Dimensions Using Weighting Functions Global-Local Orthogonal Approximation in 1-, 2-, and N-Dimensional Spaces Algorithm Implementation Properties of GLO-MAP Approximation Illustrative Engineering Applications Nonlinear System Identification Problem Statement and Background Novel System Identification Algorithm Nonlinear System Identification Algorithm Numerical Simulation Distributed Parameter Systems MLPG-Moving Least Squares Approach Partition of Unity Finite Element Method Control Distribution for Over-Actuated Systems Problem Statement and Background Control Distribution Functions Hierarchical Control Distribution Algorithm Numerical Results Appendix References Index Each chapter contains an Introduction and a Summary.

Abstract: We prove results for the interlacing of zeros of Jacobi polynomials of the same or adjacent degree as one or both of the parameters are shifted continuously within a certain range. Numerical examples are given to illustrate situations where interlacing fails to occur.

Abstract: By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials, Constructive Approximation, 11, 181–226, then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey–Wilson operator , following pioneering work by Magnus 1988, Associated Askey–Wilson polynomials as Laguerre–Hahn orthogonal polyno...

Abstract: This paper focuses on the computation of statistical moments of strains and stresses in a random system model where uncertainty is modeled by a stochastic finite element method based on the polynomial chaos expansion. It identifies the cases where this objective can be achieved by analytical means using the orthogonality property of the chaos polynomials and those where it requires a numerical integration technique. To this effect, the applicability and efficiency of several numerical integration schemes are considered. These include the Gauss–Hermite quadrature with the direct tensor product—also known as the Kronecker product—Smolyak's approximation of such a tensor product, Monte Carlo sampling, and the Latin Hypercube sampling method. An algorithm for reducing the dimensionality of integration under a direct tensor product is also explored for optimizing the computational cost and complexity. The convergence rate and algorithmic complexity of all of these methods are discussed and illustrated with the non-deterministic linear stress analysis of a plate. Copyright © 2007 John Wiley & Sons, Ltd.