Nonsymmetric interpolation macdonald polynomials and \mathfrak{g}\mathfrak{l}_n basic hypergeometric series

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type . Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for series. *Supported by the ANR project MARS (BLAN06-2 134516). **Supported by the NSF grant DMS-0401387. ***Supported by the Australian Research Council.     

Some characterizations of appell andℊ-appell polynomials

Summary Several characterizations are given for the wellknown Appell polynomials and for their basic analogues: the -Appell polynomials defined by Equation (3.3)below. The main results contained in Theorems 1, 2and 3of the present paper, and the applications considered in Section 2,are believed to be new. Some interesting connections with earlier results are also indicated.Supported, in part, by NSERC (Canada) grant A-7353.     

Inequalities for generalized nonnegative polynomials

We define generalized polynomials as products of polynomials raised to positive real powers. The generalized degree can be defined in a natural way. We prove Markov-, Bernstein-, and Remez-type inequalities inL ^p (0p) and Nikolskii-type inequalities for such generalized polynomials. Our results extend the corresponding inequalities for ordinary polynomials.Communicated by George G. Lorentz.     

Computing the Invariant Polynomials of a Polynomial Matrix. II

The paper considers the problem of computing the invariant polynomials of a general (regular or singular) one-parameter polynomial matrix. Two new direct methods for computing invariant polynomials, based on the W and V rank-factorization methods, are suggested. Each of the methods may be regarded as a method for successively exhausting roots of invariant polynomials from the matrix spectrum. Application of the methods to computing adjoint matrices for regular polynomial matrices, to finding the canonical decomposition into a product of regular matrices such that the characteristic polynomial of each of them coincides with the corresponding invariant polynomial, and to computing matrix eigenvectors associated with roots of its invariant polynomials are considered. Bibliography: 5 titles.     

Dedekind–Carlitz polynomials as lattice-point enumerators in rational polyhedra

We study higher-dimensional analogs of the Dedekind–Carlitz polynomials , where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind–Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for Dedekind–Carlitz polynomials, a characterization of Dedekind–Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell–Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes. Research of Haase supported by DFG Emmy Noether fellowship HA 4383/1. We thank Robin Chapman, Eric Mortenson, and an anonymous referee for helpful comments.     

New methods for generating permutation polynomials over finite fields

We present two methods for generating linearized permutation polynomials over an extension of a finite field Fq. These polynomials are parameterized by an element of the extension field and are permutation polynomials for all nonzero values of the element. For the case of the extension degree being odd and the size of the ground field satisfying , these parameterized linearized permutation polynomials can be used to derive non-parameterized nonlinear permutation polynomials via a recent result of Ding et al.     

Spectral methods with sparse matrices

Summary Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N ⁴) for polynomials of degree N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N ²). Certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.     

Kazhdan-Lusztig polynomials and canonical basis

In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the groupS _n coincide with the coefficients of the canonical basis innth tensor power of the fundamental representation of the quantum groupU _{q k} . We also use known results about canonical bases forU _{q 2} to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.     

Analysis of the combined finite element and Fourier interpolation

Summary We consider Lagrange interpolation involving trigonometric polynomials of degree N in one space direction, and piecewise polynomials over a finite element decomposition of mesh size h in the other space directions. We provide error estimates in non-isotropic Sobolev norms, depending additively on the parametersh andN. An application to the convergence analysis of an elliptic problem, with some numerical results, is given.     

The Fourth Order Difference Equation for the Laguerre-Hahn Polynomials Orthogonal on Special Non-uniform Lattices

We firstly establish the fourth order difference equation satisfied by the Laguerre-Hahn polynomials orthogonal on special non-uniform lattices in general case, secondly give it explicitly for the cases of polynomials r-associated to the classical polynomials orthogonal on linear, q-linear and q-nonlinear (Askey-Wilson) lattices, and thirdly give it semi-explicitly for the class one Laguerre-Hahn polynomials orthogonal on linear lattice.     

A General Algorithm for the MacMahon Omega Operator

In his famous book Combinatory Analysis MacMahon introduced Partition Analysis (Omega Calculus) as a computational method for solving problems in connection with linear diophantine inequalities and equations. The technique has recently been given a new life by G.E. Andrews and his coauthors, who had the idea of marrying it with the tools of to-days Computer Algebra.The theory consists of evaluating a certain type of rational function of the form A()^-1 B(1/)^-1 by the MacMahon operator. So far, the case where the two polynomials A and B are factorized as products of polynomials with two terms has been studied in details. In this paper we study the case of arbitrary polynomials A and B. We obtain an algorithm for evaluating the operator using the coefficients of those polynomials without knowing their roots. Since the program efficiency is a persisting problem in several-variable polynomial Calculus, we did our best to make the algorithm as fast as possible. As an application, we derive new combinatorial identities.AMS Subject Classification: 05A17, 05A19, 05E05, 15A15, 68W30.     

A simple technique for computing theH ∞-norm of polynomials

In this note, computation of theH -norm of polynomials is considered. It is shown that direct computation of theH -norm of polynomials, based on the definition of the norm, results in a simple and inexpensive technique for computing the norm.     

To solving multiparameter problems of algebra 3. Cylindrical manifolds of the regular spectrum of a matrix

Methods for computing polynomials (complete polynomials) whose zeros form cylindrical manifolds of the regular spectrum of a q-parameter polynomial matrix in the space ^q are considered. Based on the method of partial relative factorization of matrices, new methods for computing cylindrical manifolds are suggested. The W and V methods, previously proposed for computing complete polynomials of q-parameter polynomial matrices whose regular spectrum is independent of one of the parameters, are extended to a wider class of matrices. Bibliography: 4 titles._______Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 108–121.     

Generic extensions and generic polynomials for semidirect products

This paper presents a generalization of a theorem of Saltman on the existence of generic extensions with group AG over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups, and it gives new bounds on the generic dimension.Generic polynomials for several small groups are constructed.     

Narayana numbers and Schur–Szeg? composition

In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers where stands for the usual binomial coefficient, i.e. . They count Dyck paths of length n and with exactly k peaks, see e.g. [R.A. Sulanke, The Narayana distribution, in: Lattice Path Combinatorics and Applications (Vienna, 1998), J. Statist. Plann. Inference 101 (1–2) (2002) 311–326 (special issue)] and they appeared recently in a number of different combinatorial situations, see for e.g. [T. Doslic, D. Syrtan, D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67–82; A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007) 2909–2924; F. Yano, H. Yoshida, Some set partitions statistics in non-crossing partitions and generating functions, Discrete Math. 307 (2007) 3147–3160]. Strangely enough Narayana polynomials also occur as limits as n→∞ of the sequences of eigenpolynomials of the Schur–Szeg? composition map sending (n−1)-tuples of polynomials of the form (x+1)ⁿ⁻¹(x+a) to their Schur–Szeg? product, see below. We present below a relation between Narayana polynomials and the classical Gegenbauer polynomials which implies, in particular, an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N_n(x)}.     

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Given a real functionf C ^2k [0,1],k 1 and the corresponding Bernstein polynomials {B _n (f)} _n we derive an asymptotic expansion formula forB _n (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanB _n (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.This work was partially supported by a grant from MURST 40.     

Explicit representations of biorthogonal polynomials

Given a parametrised weight function (x,) such that the quotients of its consecutive moments are Möbius maps, it is possible to express the underlying biorthogonal polynomials in a closed form [5]. In the present paper we address ourselves to two related issues. Firstly, we demonstrate that, subject to additional assumptions, every such obeys (inx) a linear differential equation whose solution is a generalized hypergeometric function. Secondly, using a generalization of standard divided differences, we present a new explicit representation of the underlying biorthogonal polynomials.     

A limiting relation between Chebyschev polynomials with a discrete argument and Legendre polynomials

Asymptotic estimates are obtained in a uniform metric and in the L _p metrics (p 2) for the difference between Chebyschev polynomials with a discrete argument and Legendre polynomials, under simultaneous passage to infinity of the degree of the polynomials and the number of lattice nodes at which the Chebyschev polynomials are defined.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 37–43, 1989.     

On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes

It is shown that the fundamental polynomials for (0, 1, …, 2m+1) Hermite–Fejér interpolation on the zeros of the Chebyshev polynomials of the first kind are non-negative for −1x1, thereby generalising a well-known property of the original Hermite–Fejér interpolation method. As an application of the result, Korovkin's 10theorem on monotone operators is used to present a new proof that the (0, 1, …, 2m+1) Hermite–Fejér interpolation polynomials offC[−1, 1], based onnChebyshev nodes, converge uniformly tofasn→∞.     

Linearization and connection coefficients of orthogonal polynomials

Let {P _n } _n ^=0/ be a system of orthogonal polynomials.Lasser [5] observed that if the linearization coefficients of {P _n } _n ^=0/ are nonnegative then each of theP _n (x) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomialsP _n can be expressed in terms ofQ _n with nonnegative coefficients, where {Q _n } _n ^=0/ is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.