Abstract: By using the method of orthogonal polynomials, we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner-Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures [as, e.g., spectral form factor, number variance, and small distance behavior of the nearest neighbor distance distribution $p(s)$] are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p(s)\ensuremath{\propto}{s}^{5/2}$ for some parameter values.

Abstract: Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line. We also derive a second-order differential equation satisfied by these polynomials. We discuss the Lie algebra generated by the generalized creation and annihilation operators. From the differential equations, Plancherel - Rotach type asymptotics are derived. Under certain conditions, stated in the text, an Airy function emerges.

Abstract: Given a positive measure Σ with gs > 1, we write Μe ℳΣ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λΣ w that minimizes the weighted logarithmic energy over the class ℳΣ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λΣ w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials.

Abstract: Orthogonal polynomials satisfying fourth order differential equations were classified by H. L. Krall [K2]. They can be obtained from very special instances of the (generalized) Laguerre and the Jacobi polynomials by the Darboux process applied to semi-infinite tridiagonal matrices [GH1]. In this paper, starting from any instance of the (generalized) Laguerre and the Jacobi polynomials, we construct a one parameter family of polynomials which are eigenfunctions of a fourth order differential operator. In general, these polynomials will satisfy a five term recursion relation. Some further examples involving higher order differential operators and recursion relations are also presented. We use a proper version of the idea of a bispectral involution first proposed by G. Wilson [W1] and then reinterpreted in the context of the Darboux transformation by other authors [KR], [BHY1]. The notion of a second order bispectral differential operator was introduced in [DG]. One says that such an operator L = −d2/dx2 + V(x) is bispectral if there exists a family of its eigenfunctions L(x, d/dx)ψ(x, k) = kψ(x, k)

Abstract: Refinements of Rogers-Ramanujan type identities by K. Alladi Ramanujan's class invariants with applications to the values of $q$-continued fractions and theta functions by B. C. Berndt, H. H. Chan, and L.-C. Zhang Elementary derivations of summation and transformation formulas for $q$-series by G. Gasper $\int ^{m/6}_{n/4} \ln \Gamma (z)dz$ by R. Wm. Gosper, Jr. On a $q$-analogue of Gauss equation and some $q$-Riccati equations by F. A. Grunbaum and L. Haine Determinant evaluations and $U(n)$ extensions of Heine's $_2\phi_1$-transformations by R. A. Gustafson and C. Krattenthaler Some generating functions for $q$-polynomials by M. E. H. Ismail, D. R. Masson, and S. K. Suslov Addition formulas for $q$-special functions by E. Koelink Special functions and $q$-commuting variables by T. H. Koornwinder Multivariable Askey-Wilson polynomials and quantum complex Grassmannians by M. Noumi, M. S. Dijkhuizen, and T. Sugitani A Mathematica $q$-analogue of Zeilberger's algorithm based on an algebraically motivated approach to $q$-hypergeometric telescoping by P. Paule and A. Riese Orthogonal polynomials in the complex plane and on the real line by W. Van Assche On orthogonal polynomials in several variables by Y. Xu Appendix I: Program list of speakers and topics by D. R. Masson Appendix II: List of participants by D. R. Masson.

Abstract: Two families (type A and type B) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri-type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type A) and the Laguerre polynomials (type B), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl.

Abstract: Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions |x 1| α 1 . . . |xd | α d on the unit sphere S d-1 in ℝ d . The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

Abstract: We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

Abstract: Applying the Coulomb fluid approach to the Hermitian random matrix ensembles, universal derivatives of the free energy for a system of N logarithmically repelling classical particles under the influence of an external confining potential are derived. It is shown that the elements of the Jacobi matrix associated with the three-term recurrence relation for a system of orthogonal polynomials can be expressed in terms of these derivatives and therefore give an interpretation of the recurrence coefficients as thermodynamic susceptibilities. This provides an algorithm for the computation of the asymptotic recurrence coefficients for a given weight function. We also show that a pair of quasilinear partial differential equations, obtained in the continuum limit of the Toda lattice, can be integrated exactly in terms of certain auxilliary functions related to the initial data, and in our formulation in terms of integrals of the logarithm of the weight function. To demonstrate this procedure we give some examples where the initial data increases along the half line. Combining identities of the theory of orthogonal polynomials and certain Coulomb fluid relations, a second-order ordinary differential equation (with coefficients determined by the Coulomb fluid density) satisfied by the polynomials is derived. We use this to prove some conjectures put forward in previous papers. We show that, if the confining potential is convex, then near the edges of the spectrum of the Jabcobi matrix, orthogonal polynomials of large degree is uniformly asymptotic to Airy function.

Abstract: The relationship is made between matrix integrals, Toda master-symmetries, Virasoro constraints and orthogonal polynomials.

Abstract: A method is developed that caters for the application of correspondence analysis to two-way contingency tables with one and two ordered sets of categories. The method involves calculating orthogonal polynomials of the type described by EMERSON (1968), and partitioning the chi-square statistic using the method described in LANCASTER (1953). The method has all the features of simple correspondence analysis, although allows for additional information about the structure and association of the data to be made by isolating location, dispersion and higher order components of the rows and columns.

Abstract: Algebra of Hankels Lanczos algorithm orthogonal polyEuclidean fugues linear nomials Pade approximation linear systems general rational interpolation wavelets.

Abstract: The information entropy is explicitly obtained for the harmonic oscil- lator and the hydrogen atom (Coulomb potential) in D dimensions (D = 1,2,3). It is shown how these entropies are related to entropies involving classical orthog- onal polynomials and the physical interpretation of this information entropy is given.

Abstract: We present a short introduction into general orthogonal polynomials in the complex plane, with special attention for the real line and the unit circle Most of the material is classical and available in di®erent textbooks (see the references for relevant literature) This introduction brings together the analysis in the complex plane, the real line and the unit circle, which should be useful for those initiating their research in this eld The emphasis is on extremal properties, location of zeros, recurrence relation, and quadrature rather than on asymptotic results 1 Orthogonal polynomials in the complex plane 11 Preliminaries Let 1 be a positive Borel measure in the complex plane and consider the Hilbert space L (1) of measurable functions A for which 2 Z 2 jA(z)j d1(z) <1: In L (1) we have the inner product 2 Z hA;Ai = A(z)A(z) d1(z); A;A 2 L (1): 2 Suppose A ; A ; A ; : : : is a system of linearly independent functions in L (1) 0 1 2 2 Quite often it is much more convenient to transform this system of linearly independent functions into another system of linearly independent functions ' ; ' ; ' ; : : : 0 1 2 such that ' is a linear combination of the n+ 1 functions A ;A ; : : : ; A and n 0 1 n Z h' ;' i = ' (z)' (z) d1(z) = 0; m6= n: n m n m This new system of functions is then said to be a system of orthogonal functions with respect to the measure 1 If moreover Z 2 2 k' k = j' (z)j d1(z) = 1; n 0; n n then ' are orthonormal functions n 1991 Mathematics Subject Classi cation Primary 42C05; Secondary 33C45 Senior Research Associate of the Belgian National Fund for Scienti c Research c °0000 American Mathematical Society 0000-0000/00 $100 + $25 per page

Abstract: Israel Prototype testing and experimentation play a key role in the development of new products. It is common practice to build a single prototype product and then test it at specified operating conditions. It is often beneficial, however, to make several variants of a prototype according to a fractional factorial design. The information obtained can be important in comparing design options and improving product performance and quality. In such experiments the response of interest is often not a single number but a performance curve over the test conditions. In this article we develop a general method for the design and analysis of prototype experiments that combines orthogonal polynomials with two-level fractional factorials. The proposed method is simple to use and has wide applicability. We explain our ideas by reference to an experiment reported by Taguchi on carbon monoxide exhaust of combustion engines. We then apply them to an experiment on a prototype fluid-flow controller.

Abstract: We present a unified approach for the construction of polynomial wavelets. Our main tool is orthogonal polynomials. With the help of their properties we devise schemes for the construction of time localized polynomial bases on bounded and unbounded subsets of the real line. Several examples illustrate the new approach.

Abstract: Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for hyperexponential integrals. Further versions of this algorithm allow the computation of recurrence and differential equations from Rodrigues type formulas and from generating functions. In particular, these algorithms can be used to compute the differential/difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations, and from their Rodrigues rperesentations and generating functions. In recent work, we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials, in terms of the coefficients of their differential/difference equations, to give an algorithm to identify the polynomial system from a given recurrence equation. In this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential/difference equation, the hupergeometric representation, and the recurrence equation is possible. The main technique is again to use texplicit formulas for structural identities of the given polynomial systems.

Abstract: In the paper we formulate and verify a difference counterpart of the Macdonald-Mehta conjecture and its generalization for the Macdonald polynomials. Namely, we determine the Fourier transforms of the polynomials multiplied by the Gaussian, which is closely connected with the new difference Harish-Chandra theory.

Abstract: 1. The 2-Toda lattice and its generic symmetries 2. A Larger class of symmetries for special initial conditions 3. Borel decomposition of Moment matrices, tau-functions and string-orthogonal polynomials 4. From string-orthogonal Polynomials to the 2-Toda lattice and the string equation 5. Virasoro constraints on two-matrix integrals

Abstract: Calogero-Sutherland models associated to the Weyl groups of type A and B with exchange terms included in the Hamiltonians systems have non-symmetric eigenfunctions, which are products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differential-difference operators introduced by the author in TAMS 1989 (311), 167-183. There is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,...,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermite-type polynomials. A complete basis of eigenfunctions of Yamamoto's B_N spin Calogero model is obtained by multiplying these polynomials by the ground state.

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 these polynomials can be generated by a de Casteljau algorithm, which is a generalization of that relating to the classical case.

Abstract: Simplified chemical kinetic schemes are a crucial prerequisite for the simulation of complex three-dimensional turbulent flows, and various methods for the generation of reduced mechanisms have been developed in the past. The method of intrinsic low-dimensional manifolds (ILDM), e.g., provides a mathematical tool for the automatic simplification of chemical kinetics, but one problem of this method is the fact that the information which comes out of the mechanism reduction procedure has to be stored for subsequent use in reacting-flow calculations. In most cases tabulation procedures are used which store the relevant data (such as reduced reaction rates) in terms of the reaction progress variables, followed by table look-up during the reacting-flow calculations. This can result in huge amounts of storage needed for the multi-dimensional tabulation. In order to overcome this problem a storage scheme is presented which is based on orthogonal polynomials. Instead of the use of small tabulation cells and local mesh refinement, the thermochemical state space is divided into a small number of coarse cells. Within these coarse cells polynomial approximations are used instead of frequently used multi-linear interpolation. This leads to a considerable decrease of needed storage. The hydrogen-oxygen system is considered as an example. Even for this small chemical system, a decrease of the needed storage requirement by a factor of 100 is obtained.