Some results on numerical divided difference formulas

The remainder estimates of numerical divided difference formula are given for the functions of lower and higher smoothness, respectively. Then several divided difference formulas with super-convergence are derived with their remainder expressions.     

Uniqueness of difference polynomials of meromorphic functions

In this article, we investigate the uniqueness problems of difference polynomials of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Shibazaki. Some examples are given to show the results in this article are best possible.     

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.     

Zeros of the sum of polynomials

It is given an upper bound for the number of simple and distinct zeros of the polynomial f+g, where f and g are relatively prime polynomials with complex coefficients.     

On the location of critical points of polynomials

Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .

     

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.     

A factorization method for q-Racah polynomials

We develop a factorization method for q-Racah polynomials. It is inspired by the approach to q-Hahn polynomials based on the q-Johnson scheme, but we do not use association scheme theory nor Gel'fand pairs but only manipulation of q-difference operators.     

On the common part of the spectrum of finite difference operators generated by systems of polynomials

In this paper, we study the spectra of finite difference operators generated by systems of multiple orthogonal polynomials and the corresponding systems of measures of Stieltjes type. We show that the common support of the orthogonality measures coincides with the intersection of the spectra of the family of finite difference operators with common collection of Weyl functions.     

Zeros and fixed points of difference operators of meromorphic functions

Let f be a transcendental meromorphic function and Δf(z) = f(z + 1) − f(z). A number of results are proved concerning the existences of zeros and fixed points of Δ f(z) and Δ f(z)/f(z) when f(z) is of order σ(f)=1. Examples show that some of the results are sharp.     

Univalent polynomials and fractional order differences of their coefficients

Using fractional order differences, we find sufficient conditions on the coefficients of certain polynomials defined on the unit disk such that the zeros of these polynomials does not lie in D.     

A logician's view of graph polynomials

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we distinguish between the graph theoretic (semantic) and the algebraic (syntactic) meaning of graph polynomials. Graph polynomials appear in the literature either as generating functions, as generalized chromatic polynomials, or as polynomials derived via determinants of adjacency or Laplacian matrices. We show that these forms are mutually incomparable, and propose a unified framework based on definability in Second Order Logic. We show that this comprises virtually all examples of graph polynomials with a fixed finite set of indeterminates. Finally we show that the location of zeros and stability of graph polynomials is not a semantic property. The paper emphasizes a model theoretic view. It gives a unified exposition of classical results in algebraic combinatorics together with new and some of our previously obtained results scattered in the graph theoretic literature.     

Formulas for zonal polynomials

New integral and differential formulas for zonal polynomials are proved. As illustrations, zonal polynomials corresponding to partitions of two parts are computed. A method is presented, based on a certain partial differential operator, for expressing an orthogonally invariant polynomial as a linear combination of zonals. Zonal polynomials are expressed as linear combinations of well-known symmetric polynomials.     

On q-orthogonal polynomials over a collection of complex origin intervals related to little q-Jacobi polynomials

We construct the sequence of orthogonal polynomials with respect to an inner product which is defined by q-integrals over a collection of intervals in the complex plane. We prove that they are connected with little q-Jacobi polynomials. For such polynomials we discuss a few representations, a recurrence relation, a difference equation, a Rodrigues-type formula and a generating function. 2000 Mathematics Subject Classification Primary—33D45, 42C05