Abstract: Preface. Introduction. Level Crossings of Stochastic Processes. Algebraic Polynomials. Trigonometric Polynomials. Orthogonal Polynomials. Hyperbolic Polynomials. Other Distributions. Complex Coefficients and Complex Roots. Bibliography.

Abstract: This paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals. Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials. The first one is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set. The second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the Bernstein polynomials. Numerical results to real-world control problems are presented showing the efficiency of both algorithms.

Abstract: We consider 3-parametric polynomialsP μ * (x; q, t, s) which replace theA n-series interpolation Macdonald polynomialsP μ * (x; q, t) for theBC n-type root system For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation Ass → ∞ the polynomialsP μ * (x; q, t, s) becomeP μ * (x; q, t) We also prove a binomial formula for 6-parametric Koornwinder polynomials

Abstract: We show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. These q-analogues of Littlewood-Richardson multiplicities coincide with those previously introduced in terms of ribbon tableaux.

Abstract: We consider the gl N -invariant Calogero—Sutherland models with N = 1, 2, 3, … in the framework of symmetric polynomials. The Hamiltonian of any such model admits a distinguished orthogonal eigenbasis characterized as the union of Yangian Gelfand—Zetlin bases of irreducible components with respect to the Yangian action on the space of states. We construct an isomorphism from the space of states into the space of symmetric Laurent polynomials that maps the eigenbasis into the basis of gl N - Jack polynomials. These polynomials are defined as specializations of Macdonald polynomials where both parameters approach an Nth primitive root of unity. As an application of this isomorphism we compute two-point dynamical spin-density and density correlation functions in the gl 2-invariant Calogero-Sutherland model at integer values of the coupling constant.

Abstract: There are examples of Calogero–Sutherland models associated to the Weyl groups of type A and B. When exchange terms are added to the Hamiltonians the systems have non-symmetric eigenfunctions, which can be expressed as 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 Trans. Am. Math. Soc. 311, 167–183 (1989). After a description of known results, particularly from the works of Baker and Forrester, and Sahi; 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. These can be expressed by using the generalized binomial coefficients. A complete basis of eigenfunctions of Yamamoto's BN spin Calogero model is obtained by multiplying these polynomials by the ground state.

Abstract: Recently many people have studied the Sobolev orthogonal polynomials, that is, polynomials which are orthogonal relative to a symmetric bilinear form $\\phi(\\cdot,\\cdot)$ defined by $$ (1.1) $\\phi(p,q) := (p,q)_N = \\sum_{k=0}^{N} \\int_{R}p^(k) (x)q^(k) (x) d\\mu_k, $$ where each $d\\mu_k$ is a signed Borel measure on the real line $R$ with finite moments of all orders. For the brief history on this subject, we refer to the survey article Ronveaux [13] and Marcellan and et al [10].

Abstract: We find four linear independent solutions of the fourth-order differential equation satisfied by the associated Jacobi polynomials. We show that the coefficients in the lowering operator for general orthogonal polynomials satisfy inhomogeneous four-term recurrence relations and derive further properties of them. In addition, we show that the associated polynomials {Pn (x)} for positive integers c satisfy a linear differential equation of order four, we identify a basis of solutions of the differential equation, and we establish similar results for co-recursive polynomials.

Abstract: We prove that several polynomials naturally arising in combinatorics are Hilbert polynomials of standard graded commutative k-algebras.

Abstract: Part 1. Bispectrality: Automorphisms of the Weyl algebra and bispectral operators by B. Bakalov, E. Horozov, and M. Yakimov Huygens' principle and the bispectral problem by Y. Berest Some bispectral musings by F. A. Grunbaum Beyond the classical orthogonal polynomials by L. Haine Bispectral operators, dual isomondromic deformations and the Riemann-Hilbert dressing method by J. Harnad Darboux transformations and the bispectral problem by A. Kasman The discrete version of the bispectral problem by F. Levstein and L. F. Matusevich Explicit formulas for the Airy and Bessel bispectral involutions in terms of Calogero-Moser pairs by M. Rothstein Bispectrality and Darboux transformations in the theory of orthogonal polynomials by V. Spiridonov, L. Vinet, and A. Zhedanov Baker-Akhiezer functions and the bispectral problem in many dimensions by A. P. Veselov Bispectral algebras of ordinary differential operators by G. Wilson The bispectral problem, rational solutions of the master symmetry flows, and bihamiltonian systems by J. P. Zubelli Part 2. Related Topics: The geometry of spinors and the multicomponent BKP and DKP hierarchies by V. Kac and J. van de Leur The Hamiltonian route to Sato Grassmannian by F. Magri Darboux transformations in associative rings and functional-difference equations by V. B. Matveev Remarks about the Calogero-Moser system and the KP equation by A. Y. Orlov Subject index.

Abstract: The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) $q$-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit $q\\to 1$ leading to multivariable Racah type polynomials. Of special interest is the situation that $q$ lies on the unit circle; in that case it is found that there exists a natural parameter domain for which the discrete orthogonality measure (which is complex in general) becomes real-valued and positive. We investigate the properties of a finite-dimensional discrete integral transform for functions over the grid, whose kernel is determined by the multivariable $q$-Racah polynomials with parameters in this positivity domain.

Abstract: Combinatorial objects called rigged configurations give rise to i>q-analogues of certain Littlewood–Richardson coefficients. The Kostka–Foulkes polynomials and two-column Macdonald–Kostka polynomials occur as special cases. Conjecturally these polynomials coincide with the Poincare polynomials of isotypic components of certain graded i>GL(i>n)-modules supported in a nilpotent conjugacy class closure in i>gl(i>n).

Abstract: Positive discrete series representations of the Lie algebra $su(1,1)$ and the quantum algebra $U_q(su(1,1))$ are considered. The diagonalization of a self-adjoint operator (the Hamiltonian) in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal polynomials. Using simple realizations of $su(1,1)$, $U_q(su(1,1))$, and their representations, these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials. The relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials, or to Poisson kernels. In particular, a group theoretical derivation of the Poisson kernel for Meixner-Pollaczak and Al-Salam--Chihara polynomials is obtained.

Abstract: We study two indeterminate Hamburger moment problems and the corresponding orthogonal polynomials. The coefficients in their recurrence relations are of exponential growth or are polynomials of degree 2. The entire functions in the Nevanlinna parametrization are found. The orthogonal polynomials with polynomial recurrence coefficients resemble the Freud polynomials with a = 1/2 . Inequalities are given for the largest zero and the asymptotic behavior of the largest zero is established.

Abstract: A class of optimal control of systems with distributed parameters is considered. The process of the systems under consideration is governed by a linear parabolic partial differential equation. By use of the modal space technique, the optimal control of a distributed parameter system is simplified into the optimal control of a linear time-invariant lumped-parameter system. Next, a direct computational method for evaluating the modal optimal control and trajectory of the linear time-invariant lumped-parameter is suggested. The method is based on using finite interpolating orthogonal polynomials to approximate modal state variables. The formulation is straightforward and convenient for digital computation. An illustrative example is given to demonstrate the advantage of this method. © 1998 John Wiley & Sons, Ltd.

Abstract: In this paper we describe a new family of polynomials which are eigenfunctions of a singular Sturm–Liouville problem on the triangle T 2 = { ( x , y ) : x ≥ 0 , y ≥ 0 , x + y 1 } . The polynomials are shown to be orthogonal over T 2 with respect to a unit weight function, and may be used in approximations which are exponentially convergent for functions which are infinitely smooth in T 2 . The zeros of the polynomials may be used in cubature formulae on T 2 .