Abstract: We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx= e Q(x) dx on the real line, where Q(x)=∑ 2m k=0 qkx k , q2m> 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. c 1999 John Wiley & Sons, Inc.

Abstract: We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zig-zag paths in random domino tilings of the Aztec diamond, and also in a certain simplified directed first-passage percolation model. We use the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of Tracy and Widom. As a limit of the Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on partitions, and using this we prove a conjecture of Baik, Deift and Johansson that under the Plancherel measure, the distribution of the lengths of the first k rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel, which we call the discrete Bessel kernel, plays an important role.

Abstract: We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that \epsilon \le (n/N) \le \lambda_{cr} - \epsilon for some \epsilon > 0, where \lambda_{cr} is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously we derive semiclassical asymptotics for the recursive coefficients of the orthogonal polynomials, and we show that these coefficients form a cycle of period two which drifts slowly with the change of the ratio n/N. The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix Riemann-Hilbert problem. As an application of the semiclassical asymptotics of the orthogonal polynomials, we prove the universality of the local distribution of eigenvalues in the matrix model with the double-well quartic interaction in the presence of two cuts.

Abstract: This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

Abstract: We present a simple approach in order to compute recursively the connection coefficients between two families of classical (discrete) orthogonal polynomials (Charlier, Meixner, Kravchuk, Hahn), i.e., the coefficients Cm(n) in the expression P n (X)= ∑ n m=0 C m (n)Q m (x) , where Pn(x) and Qm(x) belong to the aforementioned class of polynomials. This is SCV2 done by adapting a general and systematic algorithm, recently developed by the authors, to the discrete classical situation. Moreover, extensions of this method allow to give new addition formulae and to estimate Cm(n)-asymptotics in limit relations between some families.

Abstract: Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t = (t(1), t(2), ...), leads to the standard Toda lattice and tau-functions, expressed as hermitian matrix integrals. This paper is concerned with a sequence of t-perturbed weights, rather than one single weight. This sequence leads to moments, polynomials and a (fuller) matrix evolving according to the discrete KP-hierarchy. The associated tau-functions have integral, as well as vertex operator representations. Among the examples considered, we mention: nested Calogero-Moser systems, concatenated solitons and m-periodic sequences of weights. The latter lead to 2m + 1-band matrices and generalized orthogonal polynomials, also arising in the context of a Riemann-Hilbert problem. We show the Riemann-Hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower-times upper-triangular matrix.

Abstract: We calculate the joint probability distribution of the Wigner-Smith time-delay matrix Q=−iℏS −1∂S/∂e and the scattering matrix S for scattering from a chaotic cavity with ideal point contacts. To this end we prove a conjecture by Wigner about the unitary invariance property of the distribution functional P[S(e)] of energy-dependent scattering matrices S(e). The distribution of the inverse of the eigenvalues τ1,…,τ N of Q is found to be the Laguerre ensemble from random-matrix theory. The eigenvalue density ρ(τ) is computed using the method of orthogonal polynomials. This general theory has applications to the thermopower, magnetoconductance, and capacitance of a quantum dot.

Abstract: Unilateral buckling is a contact problem whereby buckling is confined to take place in only one lateral direction. For plate structures, this can occur when a thin steel plate is juxtaposed with a rigid concrete medium and the steel may only buckle locally away from the concrete core. This paper investigates the use of simple and orthogonal polynomials in the Rayleigh–Ritz method for unilateral plate buckling. The orthogonal polynomials used are the classical Chebyshev types 1 and 2, Legrende, Hermite and Laguerre. The study presents a comparison between the efficiency of the polynomial-based displacement functions with regard to elastic bilateral and unilateral plate buckling, where efficiency is measured as a function of their convergence characteristics. Some buckling solutions for plates with varying boundary conditions and in-plane shear loads are also provided as an illustration. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: First, T-polynomials, which were investigated in Part I, are used for a complete description of minimal polynomials on two intervals, of Zolotarev polynomials, and of polynomials minimal under certain constraints as Schur polynomials or Richardson polynomials. Then, based on an approach of W. J. Kammerer, it is shown that there exists a T-polynomial on a set of l intervals El if l + 1 boundary points of El and the number of extremal points in each interval of El are given. Finally, a fast algorithm for the numerical computation is provided and for two intervals it is demonstrated how to get T-polynomials with the help of Grobner bases.

Abstract: Science fiction and Macdonald's polynomials by F. Bergeron and A. M. Garsia On the expansion of elliptic functions and applications by R. Chouikha Generalized hypergeometric functions-Classification of identities and explicit rational approximations by D. V. Chudnovsky and G. V. Chudnovsky Tensor products of $q$-superalgebras and $q$-series identities. I by W. S. Chung, E. G. Kalnins, and W. Miller, Jr. $q$-Racah polynomials for $BC$ type root systems by J. F. van Diejen and J. V. Stokman Intertwining operators of type $B_N$ by C. F. Dunkl Symmetries and continuous $q$-orthogonal polynomials by R. Floreanini, J. LeTourneux, and L. Vinet Addition theorems for spherical polynomials on a family of quantum spheres by P. G. A. Floris On a $q$-analogue of the string equation and a generalization of the classical orthogonal polynomials by F. A. Grunbaum and L. Haine The $q$-Bessel function on a $q$-quadratic grid by M. E. H. Ismail, D. R. Masson, and S. K. Suslov Three statistics on lattice paths by D. Kim and D. Stanton Quantum Grothendieck polynomials by A. N. Kirillov $q$-difference raising operators for Macdonald polynomials and the integrality of transition coefficients by A. N. Kirillov and M. Noumi Great powers of $q$-calculus by B. A. Kupershmidt $q$-special functions: Differential-difference equations, roots of unity, and all that by V. Spiridonov On algebras of creation and annihilation operators by A. Strasburger.

Abstract: The behavior of the Lagrange polynomial L m (w,f) , based on the zeros of the orthogonal polynomials, is studied in some weighted Besov spaces B p r,q (u) . It is proved that L m (w) is a uniformly bounded map under suitable conditions on the weight functions and the parameters p , r , and q .

Abstract: We describe an alternative approach to some results of Vassiliev on spaces of polynomials, by using the scanning method which was used by Segal in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.

Abstract: Laguerre 2D polynomials are defined and their properties are investigated. The Laguerre 2D functions, introduced in [1, 2] are related to the Laguerre 2D polynomials in such a way that they also include the weight function for the orthonormalization of the Laguerre 2D polynomials. A one-parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied, in particular, to Hermite polynomials and to Laguerre 2D polynomials. These transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments. They contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for Hermite and Laguerre 2D polynomials. Applications of the Laguerre 2D polynomials and 2D functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the Fock-state basis and by normally and otherwise ordered moments are considered. The inversion of these representations is obtained in all cases. A restricted design of quasiprobabilities should become possible.

Abstract: Method and system for determining the effects of variation of N statistically distributed variables x1,...,xN (N≥1) in a fabrication process for semiconductor and other electronic devices by constructing and using an N-variable model function G(x1,...,xN) to model the process. A sequence of orthogonal polynomials Hi,ri(xi) is associated with each probability density function pi(xi) for each variable xi. These orthogonal polynomials, and products of these polynomials, are used to construct the model function G(x1,...,xN), having undetermined coefficients. Coefficient values are estimated by results of measurements or simulations with variable input values determined by the zeroes (collocation points) of selected orthogonal polynomials. A Monte Carlo process is applied to estimate a probability density function associated with the process or device. Coefficients whose magnitudes are very small are used to identify regions of (x1,...,xN)-space where subsequent Monte Carlo sampling may be substantially reduced. Monte Carlo sampling for the process is also made more transparent using the function G(x1,...,xN). The function G(x1,...,xN) can be used to calculate individual and joint statistical moments for the process.

Abstract: A cell decomposition of the space of polynomials of least deviation from zero on a system of several closed intervals of the real axis is discussed. An effective method for calculating the in each cell making use of automorphic functions is put forward.

Abstract: A Chebyshev polynomial of a square matrix A is a monic polynomial p of specified degree that minimizes |p (A)|2. The study of such polynomials is motivated by the analysis of Krylov subspace iterations in numerical linear algebra. An algorithm is presented for computing these polynomials based on reduction to a semidefinite program which is then solved by a primal-dual interior point method. Examples of Chebyshev polynomials of matrices are presented, and it is noted that if A is far from normal, the lemniscates of these polynomials tend to approximate pseudospectra of A.

Abstract: We report about results revolving around Kostka-Foulkes and parabolic Kostka polynomials and their connections with Representation Theory and Combinatorics. It appears that the set of all parabolic Kostka polynomials forms a semigroup, which we call {\it Liskova semigroup}. We show that polynomials frequently appearing in Representation Theory and Combinatorics belong to the Liskova semigroup. Among such polynomials we study rectangular $q$-Catalan numbers; generalized exponents polynomials; principal specializations of the internal product of Schur functions; generalized $q$-Gaussian polynomials; parabolic Kostant partition function and its $q$-analog; certain generating functions on the set of transportation matrices. In each case we apply rigged configurations technique to obtain some interesting and new information about Kostka-Foulkes and parabolic Kostka polynomials, Kostant partition function, MacMahon, Gelfand-Tsetlin and Chan-Robbins polytopes. We describe certain connections between generalized saturation and Fulton's conjectures and parabolic Kostka polynomials; domino tableaux and rigged configurations. We study also some properties of $l$-restricted generalized exponents and the stable behaviour of certain Kostka-Foulkes polynomials.

Abstract: Orthogonal polynomials and their properties convergence and summability of Fourier series in L2m Fourier orthogonal series in Lrm and C Fourier polynomial series in L1m - analogues of Fatou theorems the representations of the trilinear kernels generalized translation operator in orthogonal polynomials.

Abstract: Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionals ω0 and ω1 (see Definition 11) Now, let {Qn}n≥0 be the sequence of polynomials defined by Qn:=(n

Abstract: In this paper, we study orthogonal polynomials with respect to the bilinear form B S (f, g) = F (c)AG(c) T + 〈u, f g〉, where u is a quasi-definite (or regular) linear functional on the linear space P of real polynomials, c is a real number, N is a positive integer number, A is a symmetric N×N real matrix such that each of its principal submatrices are regular, and F (c) = (f(c), f ′(c), . . . , f (N−1)(c)), G(c) = (g(c), g′(c), . . . , g(N−1)(c)). For these non–standard orthogonal polynomials, algebraic and differential properties are obtained, as well as their representation in terms of the standard orthogonal polynomials associated with u. Running title: Discrete–continuous Sobolev polynomials, 1991 Mathematics Subject Classification: 33C45, 42C05

Abstract: We study asymptotics for orthogonal polynomials and other extremal polynomials on infinite discrete sets, typical examples being the Meixner polynomials and the Charlier polynomials. Following ideas of Rakhmanov, Dragnev and Saff, weshow that the asymptotic behaviour is governed by a constrained extremal energy problem for logarithmic potentials, which can be solved explicitly. We give formulas for the contracted zero distributions, the th root asymptotics and the asymptotics of the largest zeros.

Abstract: The Lagrange-mesh numerical method has the simplicity of a mesh calculation and the accuracy of a variational calculation. A flexible general procedure for deriving an infinity of new Lagrange meshes related to orthogonal or nonorthogonal bases is introduced by using nonclassical orthogonal polynomials. As an application, different Lagrange meshes based on shifted Gaussian functions are constructed. A simple quantum-mechanical example shows that the Lagrange-mesh method may become more accurate than the original variational calculation with a nonorthogonal basis.