Abstract: We extend the definition of the classical Jacobi polynomials withindexes ?, β>?1 to allow ? and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.

Abstract: Introduction and Orientation I. Symbolic computations: *Remarks *Manipulation of polynomials *Manipulations of rational functions of polynomials *Manipulations of trigonometric expressions *Systems of linear and nonlinear equations *Classical analysis *Differential equations *Integral transforms and generalized functions *Three applications *Overview II Classical orthogonal polynomials: *Remarks *General properties of orthogonal polynomials *Hermite polynomials *Jacobi polynomials *Gegenbauer polynomials *Laguerre polynomials *Legendre polynomials *Chebyshev polynomials T *Chebyshev polynomials U *Relationships among the orthogonal polynomials *Overview III Classical special functions: *Remarks/Introduction *Gamma, beta, and polygamma functions *Error functions and Fresnel integrals *Sine, cosine, exponential, and logarithmic integral functions *Bessel and airy functions *Legendre functions *Hypergeometric functions *Elliptic integrals *Elliptic functions *ProductLog function *Mathieu functions * Additional special functions *Solution of quintics with hypergeometric functions *Overview Index

Abstract: Dedicated to Professor Laura Gori on her 70th birthday Abstract: Six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli (2) ,L.Euler (4) ,E.Lucas (8) ,P. E. Appell (1) ,A. H¨ urwitz (6) and D. H. Lehmer (7) .I nthis note we deal with a new deter- minantal definition for Bernoulli polynomials recently proposed by F. Costabile (3); in particular, we emphasize some consequent procedures for automatic calculation and re- cover the better known properties of these polynomials from this new definition. Finally, after we have observed the equivalence of all considered approaches, we conclude with a circular theorem that emphasizes the direct equivalence of three of previous approaches.

Abstract: The quantum version of a non-linear oscillator, previouly analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form $m={(1+\lambda x^2)}^{-1}$ and with a $\la$-dependent nonpolynomial rational potential. This $\la$-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for $\la\to 0$ all the characteristics of the linear oscillator are recovered. Firstly, the $\la$-dependent Schr\"odinger equation is exactly solved as a Sturm-Liouville problem and the $\la$-dependent eigenenergies and eigenfunctions are obtained for both $\la>0$ and $\la<0$. The $\la$-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as $\la$-deformations of the standard Hermite polynomials. In the second part, the $\la$-dependent Schr\"odinger equation is solved by using the Schr\"odinger factorization method, the theory of intertwined Hamiltonians and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a $\la$-dependent Rodrigues formula, a generating function and $\la$-dependent recursion relations between polynomials of different orders.

Abstract: The analytic solutions of the one-dimensional Schrodinger equation for the trigonometric Rosen–Morse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention to the fact that the complex Jacobi polynomials have non-trivial orthogonality properties which make them uncomfortable for physics applications. Instead we here solve the above equation in terms of real orthogonal polynomials. The new solutions are used in the construction of the quantum-mechanical superpotential.

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 discuss several other results based on the properties of the polynomials: the equivalence between the Stieltjes-Wigert matrix model and the discrete model that appears in q-2D Yang-Mills and the relationship with Rogers-Szego polynomials and the corresponding equivalence with an unitary matrix model. Finally, we also give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

Abstract: The purpose of this paper is to construct q-Euler numbers and polynomials by using p-adic q-integral equations on Zp. Finally, we will give some interesting formulae related to these q-Euler numbers and polynomials.

Abstract: Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large n. Motivated by two plots, we look at the finer structure for the case of the Verblunsky coefficients and for what we call the BLS condition: αn = Cb^n + O ((bΔ)^n). In the former case, we describe the results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros.

Abstract: A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in detail. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions between the two dual bases are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order q-difference equations which generalize those associated with the Askey–Wilson orthogonal polynomials with a discrete argument. Normalizing the fundamental solution to unity, the hierarchies of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples. The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given.

Abstract: We provide a very general result which identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover within a unified framework the HVZ Theorem and Krein's results on orthogonal polynomials with finite essential spectra.

Abstract: We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.

Abstract: We study polynomials that are orthogonal with respect to a varying quartic weight \exp(-N(x^2/2+tx^4/4)) for t<0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its, and Kitaev, showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painleve I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric combinations of contours. Special features in the steepest descent analysis are a modified equililbrium problem and the use of Psi-functions for the Painleve I equation in the construction of the local parametrix.

Abstract: We give a combinatorial formula for the non-symmetric Macdonald polynomials E_{\mu}(x;q,t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J_{\mu}(x;q,t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.

Abstract: We describe the asymptotic behavior of the multivariate BC-type Ja- cobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results describe the approximation of the spherical functions of the infinite-dimensional symmetric spaces of type B, C, D or BC by the spherical functions of the corresponding finite-dimensional symmetric spaces. Similar results for the Jack polynomials were established in our earlier paper (Intern. Math. Res. Notices 1998, no. 13, 641-682; arXiv: q-alg/9709011). The main results of the present paper were obtained in 1997.

Abstract: Orthogonal Polynomials, Quadrature, and Approximation: Computational Methods and Software (in Matlab).- Equilibrium Problems of Potential Theory in the Complex Plane.- Discrete Orthogonal Polynomials and Superlinear Convergence of Krylov Subspace Methods in Numerical Linear Algebra.- Orthogonal Rational Functions on the Unit Circle: from the Scalar to the Matrix Case.- Orthogonal Polynomials and Separation of Variables.- An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials.- Painleve Equations - Nonlinear Special Functions.

Abstract: In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some new directions for a future research are formulated.

Abstract: In this paper, we present a systematic investigation of a unification (and generalization) of the Chan–Chyan–Srivastava multivariable polynomials and the multivariable extension of the familiar Lagrange–Hermite polynomials. We derive various classes of multilinear and mixed multilateral generating functions for these unified polynomials. We also discuss other miscellaneous properties of these general families of multivariable polynomials.

Abstract: In this paper, we define a new q–analogy of the Bernoulli polynomials and the Bernoulli numbers and we deduced some important relations of them. Also, we deduced a q–analogy of the Euler-Maclaurin formulas. Finally, we present a relation between the q–gamma function and the q–Bernoulli polynomials.

Abstract: Questions on random matrices and on non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The purpose of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of time deformations for each weight, the multi-component KP-hierarchy: these determinants are thus "tau-functions" for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. As an application, the multi-component KP-hierarchy leads to a large set of non-linear PDE's, which are useful in finding partial differential equations for the transition probabilities of certain infinite-dimensional diffusions.

Abstract: A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order $q-$difference equations which generalize the ones associated with the Askey-Wilson orthogonal polynomials with a discrete argument Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given

Abstract: We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products of Grothendieck polynomials in each set of variables, with coefficients Schubert structure constants for Grothendieck polynomials. The other type is in terms of chains in the Bruhat order. We compare this second type to other constructions of Grothendieck polynomials within the more general context of double Grothendieck polynomials and the closely related H-polynomials. Our methods are based upon the geometry of permutation patterns.

Abstract: Connections between Toda lattices (Toda chains) and similar non-linear chains and the theory of orthogonal polynomials on the real axis have been studied in detail during the last decades. Another system of difference differential equations, known as the Schur flow, is considered in this paper in the framework of the theory of orthogonal polynomials on the unit circle. A Lax pair for this system is found and the dynamics of the corresponding spectral measure is described. The general result is illustrated by an example of Bessel modified measures on the unit circle: the large-time asymptotic behaviour of their reflection coefficients is determined.

Abstract: This article focuses on those problems about extremal properties of polynomials that were considered by the Hungarian mathematicians Lipot Fejer, Mihaly Fekete, Marcel Riesz, Alfred Renyi, Gyorgy Polya, Gabor Szegő, Pal Erdős, Pal Turan, Geza Freud, Gabor Somorjai, and their associates, who lived and died mostly in the twentieth century. It reflects my personal taste and is far from complete even within the subdomains we focus on most, namely inequalities for polynomials with constraints, Muntz polynomials, and the geometry of polynomials. There are separate chapters of this book devoted to orthogonal polynomials, interpolation, and function series, so here we touch these issues only marginally.