关于复差分微分理论的若干结果

本文主要利用差分的Nevanlinna理论,研究了几种不同类型的复差分微分多项式的零点情况,推广了微分多项式理论中的一些经典结果,同时也推广了部分差分多项式的结果.另外,本文还得到了某些差分微分方程解的存在性.     

关于超越亚纯函数的一类复微分-差分多项式的零点

主要运用Nevanlinna值分布理论,研究了一类关于超越亚纯函数的复差分-微分多项式的零点问题,推广了差分-微分多项式的一些结果.利用分析函数的零点与极点的方法,证明了n取一定值时,复差分-微分多项式取零点无穷多次,结果可被看作Hayman猜想的微分-差分形式.     

有限级超越整函数的(微-)差分多项式的零点分布

作者研究了有限级超越整函数的差分多项式和微-差分多项式的零点分布,在一定条件下得到了这些多项式的零点收敛指数的精确估计.所得结果可视为Hayman关于Picard例外值的经典结果的(微-)差分模拟.     

关于Hayman问题的差分模拟的值分布

本文研究了Hayman问题的一些经典结果的差分模拟问题.利用Nevanlinna理论,获得了一类差分多项式零点密指量下界的精确估计,改进了已有的一些结论.     

高阶q-差分多项式的值分布

研究了高阶q-差分多项式的值分布性质.特别地,利用Nevanlinna理论考虑了差分多项式f(z)~n△_q~kf(z)-a(z)及其导数的零点分布,其中q∈C\{0,1}是使得△_q~kf(z)■0的常数,a(z)(■0,∞)是f(z)的小函数.     

亚纯函数的差分多项式

假设函数f(z)是亚纯函数,H(z,f)是关于f(z)的差分多项式,s(z)是关于f(z)的小函数,考察了差分多项式f(z)~nH(z,f)-s(z)的零点分布问题.首先得到了差分多项式f(z)~nH(z,f)-s(z)的零点计数函数和函数f(z)的特征函数以及极点计数函数之间的一些不等式估计,再根据这些不等式,建立了Hayman关于亚纯函数的一个经典结果的差分模拟.     

四元数体上两类多项式的零点

本文研究四元数体 Q上多项式的零点 ,特别对于其中两类多项式——系数两两可换的多项式和二次多项式建立了系统而完善的零点理论 .     

非线性微分-差分方程的解

本文研究了非线性微分-差分方程f(z)~n+a_(n-1)f(z)~(n-1)+…+a_1f(z)+q(z)e~(Q(z))f~((k))(z+c)=P(z)的有穷级非零整函数解的增长性和零点分布问题.利用微分-差分Nevanlinna值分布的方法,获得了当方程的系数满足一定条件时,方程解的增长性估计和零点分类.特别地,当n=2, a_1≠0指数多项式解满足某些条件时,获得了解具有特别的形式.该结果推广了先前文献[1,2]的结果.     

关于微分多项式零点的定量估计

近来有不少各种有关超越整函数或满足δ(∞,f)=1的亚纯函数f的特殊形式(微分多项式)存在有无穷多个零点或亏值的结果.本文将有些结果归结推广到对任意一超越亚纯函数的一类具广泛形式的微分多项式的零点作讨论,并得到定量的估计.在证明中用到了改进的有关微分多项式的Clunie引理并对各种辅助函数的零点重数作了较精密的估计.     

关于一类亚纯函数差分多项式的值分布

研究一类超级σ2(f)1的亚纯函数差分多项式的零点分布,改进了文献[1-2]的一些成果,也研究了分担小函数的超越整函数的唯一性问题.     

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.     

Uniqueness and zeros of <Emphasis Type="Italic">q</Emphasis>-shift difference polynomials

In this paper, we consider the zero distributions of q-shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to q-shift difference polynomials. We also investigate the uniqueness problem of q-shift difference polynomials that share a common value.     

Some results on zeros and uniqueness of difference-differential polynomials

We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions.We also investigate the uniqueness problems of difference...     

Random block matrices generalizing the classical Jacobi and Laguerre ensembles

In this paper we consider random block matrices which generalize the classical Laguerre ensemble and the Jacobi ensemble. We show that the random eigenvalues of the matrices can be uniformly approximated by the zeros of matrix orthogonal polynomials and obtain a rate for the maximum difference between the eigenvalues and the zeros. This relation between the random block matrices and matrix orthogonal polynomials allows a derivation of the asymptotic spectral distribution of the matrices.     

On the log-convexity of combinatorial sequences

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for certain classes of graphs, while showing that they are false in general.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

Applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems

We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball.     

The Fox-Wright functions and Laguerre multiplier sequences

Linear operators which (1) preserve the reality of zeros of polynomials having only real zeros and (2) map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain Fox-Wright functions and generalized Mittag-Leffler functions. The paper includes several open problems and questions.     

Quasi-orthogonality of some hypergeometric polynomials

The zeros of quasi-orthogonal polynomials play a key role in applications in areas such as interpolation theory, Gauss-type quadrature formulas, rational approximation and electrostatics. We extend previous results on the quasi-orthogonality of Jacobi polynomials and discuss the quasi-orthogonality of Meixner–Pollaczek, Hahn, Dual-Hahn and Continuous Dual-Hahn polynomials using a characterization of quasi-orthogonality due to Shohat. Of particular interest are the Meixner–Pollaczek polynomials whose linear combinations only exhibit quasi-orthogonality of even order. In some cases, we also investigate the location of the zeros of these polynomials for quasi-orthogonality of order 1 and 2 with respect to the end points of the interval of orthogonality, as well as with respect to the zeros of different polynomials in the same orthogonal sequence.     

Krawtchouk多项式零点的渐近展开及其误差限

Krawtchouk多项式在现代物理学中有着广泛应用．基于Li和Wong的结果,利用Airy函数改进了Krawtchouk多项式的渐近展开式,而且得到了一个一致有效的渐近展开式A·D2进一步,利用Airy函数零点的性质,推导出了Krawtchouk多项式零点的渐近展开式,并讨论了其相应的误差限．同时还给出了Krawtchouk多项式和其零点的渐近性态,它优于Li和Wong的结果．