On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices

In this article,we obtain a new fundamental theorems for Nikiforov-Uvarov-Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation,its expression is different from Suslov’s Theorem.We also establish the adjoint equation for Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on non-uniform lattices,and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well.The particular solutions of the adjoint equation are then obtained.As an appliction of these particular solutions,we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices and other important results.     

Difference Equations and Discriminants for Discrete Orthogonal Polynomials

We prove that any set of polynomials orthogonal with respect to a discrete measure supported on equidistant points contained in a half line satisfy a second order difference equation. We also give a discrete analogue of the discriminant and give a general formula for the discrete discriminant of a discrete orthogonal polynomial. As an application we give explicit evaluations of the discrete discriminants of the Meixner and the Hahn polynomials. A difference analogue of the Bethe Ansatz equations is also mentioned.Research partially supported by NSF grant DMS 99-70865     

The H q -Classical Orthogonal Polynomials

This is an expository paper that aims to give an essentially self-contained survey of a so-called q-analogue of classical polynomials from their characterizations by Hahn's property and a Rodrigues formula which allows us to construct them. Integral representations of corresponding forms are given, except in some cases where the problem remains open.     

The Christoffel Function for Orthogonal Polynomials on a Circular Arc

For the special type of weight functions on circular arc we study the asymptotic behavior of the Christoffel kernel off the arc and of the Christoffel function inside the arc. We prove Totik's conjecture for the Christoffel function corresponding to such weight functions.     

正交多项式及Pade逼近

利用Ｌｅｇｅｎｄｒｅ多项式的性质，得到ｅｘｐ（ｘ），ｔａｎｘ和ｔａｎｈｘ简单形式的对角Ｐａｄｅ逼近，在［－１，１］上Ｐｎ（ｘ）对于任意较低次幂的多项式是正交的·在求得某些函数的分母时，利用了Ｇａｕｓ求积公式·     

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.     

Strong Asymptotics for the Continuous Sobolev Orthogonal Polynomials on the Unit Circle

Strong (or Szeg -type) asymptotics for orthogonal polynomials with respect to a Sobolev inner product with general measures (the first measure is arbitrary and the second one is absolutely continuous and satisfying a smoothness condition) is obtained. Examples, illustrating the theorems proved, are presented.     

Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients tend to some complex number a with 0<a<1. The orthogonality measure μ then lives essentially on the arc {e^it :αt2π−α} where sin with α(0,π). Under the certain rate of convergence it was proved in (Golinskii et al. (J. Approx. Theory96 (1999), 1–32)) that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients.     

Generalized Bernstein Polynomials

We introduce polynomials B ⁿ _i (x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B ⁿ _i (x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Continuous and Discrete Tight Frames of Orthogonal Polynomials for a Radially Symmetric Weight

This paper considers tight frame decompositions of the Hilbert space ℘ _n of orthogonal polynomials of degree n for a radially symmetric weight on ℝ ^d , e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p∈℘ _n with the property that each f∈℘ _n can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials p _ξ obtained from p by moving its pole to ξ∈S:={ξ∈ℝ ^d :|ξ|=1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for ℘ _n .      

On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation,Factorization and Solutions

We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.     

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L²( ) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with lim_n→∞ ∑ⁿ⁻¹_k=0 |a_k|=0, {a_n}_n0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|_n|² dσ}_n0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szeg class which consists of measures with ∑^∞_n=0 (1−|a_n|²)^1/2<∞ and describe it in terms of Hessenberg matrices.     

Orthogonal Polynomials on the Ball and the Simplex for Weight Functions with Reflection Symmetries

Generalized classical orthogonal polynomials on the unit ball B ^d and the standard simplex T ^d are orthogonal with respect to weight functions that are reflection-invariant on B ^d and, after a composition, on T ^d , respectively. They are also eigenfunctions of a second-order differential—difference operator that is closely related to Dunkl's h -Laplacian for the reflection groups. Under a proper limit, the generalized classical orthogonal polynomials on B ^d converge to the generalized Hermite polynomials on R ^d , and those on T ^d converge to the generalized Laguerre polynomials on R ^d ₊ . The latter two are related to the Calogero—Sutherland models associated to the Weyl groups of type A and type B . February 14, 2000. Date revised: July 26, 2000. Date accepted: August 4, 2000.     

The Bailey Lemma and Kostka Polynomials

Using the theory of Kostka polynomials, we prove an A _n–1 version of Bailey's lemma at integral level. Exploiting a new, conjectural expansion for Kostka numbers, this is then generalized to fractional levels, leading to a new expression for admissible characters of A⁽¹⁾ _n–1 and to identities for A-type branching functions.     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].     

A Semi-Conjugate Matrix Boundary Value Problem for General Orthogonal Polynomials on an Arbitrary Smooth Jordan Curve

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear in the theory of Riemann -Hilbert approach to asymptotic analysis for orthogonal polynomials on a real interval introduced by Fokas, Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.     

On the p-Ranks and Characteristic Polynomials of Cyclic Difference Sets

In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth–Kumar–Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach.     

单位圆周上正交多项式渐近分析的Riemann-Hilbert方法

在单位圆周上考虑带特定权函数的正交多项式,利用Deift P．和Zhou X．所引进的关于振荡型Riemann-Hilbert问题的最速下降法,建立了这类正交多项式在整个复平面上的强渐近公式,发展和改进了一些经典结果．     

Ratio Asymptotics for Polynomials Orthogonal on Arcs of the Unit Circle

Ratio asymptotics for orthogonal polynomials on the unit circle is characterized in terms of the existence of lim _n |Φ _n (0)| and {lim _n [ Φ _n+1 (0)/ Φ _n (0)] , where denotes the sequence of reflection coefficients. The limit periodic case, that is, when these limits exist for n = j mod k , j = 1, . . ., k , is also considered. December 27, 1996. Date revised: October 14, 1997. Date accepted: December 22, 1997.