Some properties of biorthogonal polynomials and their application to Padé approximations

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders [N?1/N],N ε ?, for the functions $$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$ wheref(z) is a function with known Padé approximants of the indicated orders,T _j [f;z] are Taylor polynomials of degreej for the functionf(z), and α _{m, M} = $\overline {1,M} $ are constants.     

Transcendency of local conjugacies in complex dynamics and transcendency of their values

Letp andq be polynomials of the same degree. A classical result of Böttcher says that there exists a functionf conformal in a neighborhood of infinity such thatf(p(z))=q(f(z)). We show thatf is transcendental and takes transcendental values at algebraic points unlessp andq are linearly conjugate to monomials or Chebychev polynomials. As an application, we show that the conformal map from the exterior of the Mandelbrot set onto the exterior of the unit disk takes transcendental values at algebraic points. A second application is the solution of a transcendency problem posed by Golomb.     

An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

亚纯函数差分的亏量与值分布

研究亚纯函数差分的亏量与值分布,证明了:设c是一个非零有穷复数,f(z)是复平面C上的超越整函数,n是一个正整数.若△_c~nf(z)≠0,则或者f(z)取每个有穷复数无穷多次,或者△_c~nf(z)取每个非零有穷复数无穷多次.     

几类亚纯函数q移动微差分多项式的零点与Nevanlinna亏量

The first purpose of this paper is to study the properties on some q-shift difference differential polynomials of meromorphic functions,some theorems about the zeros of some q-shift difference-differential polynomials with more general forms are obtained.The second purpose of this paper is to investigate the properties on the Nevanlinna deficiencies for q-shift difference differential monomials of meromorphic functions,we obtain some relations amongδ(∞,f),δ(∞,f′),δ(∞,f(z)ⁿf(qz+c)^mf′(z)),δ(∞,f(qz+c);f′(z))andδ(∞,f(z)ⁿf(qz+c)^m).     

The distribution of extreme points in best complex polynomial approximation

LetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if $$A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,$$ wherep _n ^* (f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencen _k with the property that the sequence of (n _k +2)-point Fekete subsets of \(A_{n_k }\) has limiting distribution (ask→∞) equal to the equilibrium distribution forK. Analogues for weighted approximation are also given.     

VALUE DISTRIBUTION OF MEROMORPHIC SOLUTIONS OF CERTAIN NON-LINEAR DIFFERENCE EQUATIONS

In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference ?f(z) and divided difference?f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.     

某类非线性微分方程超越亚纯解的进一步结果

本文研究非线性微分方程f~n+Q_d(z,f)=P_1(z)e~(α_1(z))+p_2(z)e~(α_2(z))超越亚纯解的存在性和形式,其中n≥4是整数,Q_d(z,f)是关于f的次数d≤n-3且系数为有理函数的微分多项式,p_1,p_2是非零的有理函数,α_1,α_2是非常数的多项式.运用Nevanlinna值分布理论,能够得到该方程存在超越亚纯解时p_1,p_2,α_1及α_2所满足的条件.特别地,还考虑了当Q_d(z,f)=a(z)ff'且n=4时方程的超越亚纯解的存在性和形式,其中a(z)是一个非零的有理函数.     

某类均差分的值分布

研究了某类均差分的值分布问题.令n,m∈N,n>m,c∈C\{0},f(z)是超越亚纯函数,P(z)为非零多项式.一系列关于均差分F(z)=(Δ_c~nf(z))/(Δ_c~mf(z))和F~*(z)=(Δ_c~nf(z))/(Δ_c~mf(z))-P(z)的零点存在性的结果被证明.     

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.     

ON PRIMALITY OF THE COMBINATION OF EXPONENTIAL FUNCTIONS

The author discusses in this paper the transcendental unsolvability of the functionalequation F(z)=fog(z) with f being meromorphic and g entire,for the function of theformwhere Q_j's are rational,P_j's are polynomials. The main results are:a) F(z) is pseudo-prime, i. e. F=fog has no transcendental solutions f and g;b)If 0≤n_1相似文献   

整函数及其差分分担有限集的唯一性

研究了整函数及其差分多项式分担有限复数集的唯一性,得到了如下结果:设S_m={1,ω,…,ω~(m-1)},其中ω=cos(2π/m)+i sin(2π/m),c为非零有限复数,n(>5),m(≥2)均为正整数.如果f(z),g(z)为有限级整函数,满足E(S_m,f(z)~n(f(z)-1)f(z+c))=E(S_m,g(z)~n(g(z)-1))g(z+c)),那么f(z)≡g(z).     

Extremal problems in a subclass of entire functions of finite power

We study the subclass W_σ(A) of the class of entire transcendental functions f(z)of exponential type with index not greater than σ satisfying the condition $$\int_{ - \infty }^\infty {\left| {f(x)} \right|^2 dx \leqslant A^2 .}$$ We find the set of values of the quantities f(z), f′(z), etc. when z is fixed and f runs through the subclass W_σ(A). We study extremal values of functionals of the type Φ(f(z), f ′(z)). In particular, we obtain upper bounds on the quantities ¦f(z +β/2) ± f(z?β/2) ¦ and ¦af '(z) + bof(z)¦.     

微分多项式的值分布

文[2-6]讨论了亚纯函数或代数体函数的微分多项式ω‘(z)-αω^n(z)及ω（z)ω^n(z)的例外值问题。本文利用代数体函数的Nevanlinna值分布理论方法，研究了代数体函数的微分多项式P（ω^(k)(z))-aΠk=1^s(ω(z)-ak)^pk,及a(w^(k)(z)^tw^n(t)的Picard值问题，推广了他们的结果。     

On a Composition Preserving Inequalities between Polynomials

The Schur-Szegö composition of two polynomials \(f\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{z^j}} \) and \(g\left( z \right) = \sum\nolimits_{j = 0}^n {{B_j}{z^j}} \), both of degree n, is defined by \(f * g\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{B_j}{{\left( {\begin{array}{*{20}{c}}n \\ j \end{array}} \right)}^{ - 1}}{z^j}} \). In this paper, we estimate the minimum and the maximum of the modulus of f * g(z) on z = 1 and thereby obtain results analogues to Bernstein type inequalities for polynomials.     

On the growth of meromorphic and entire solutions of second-order algebraic differential equations

In this paper, the growth of the meromorphic solutions of the equation

Kernel of the second order Cauchy difference on groups

Let (G, ·) be a group, (H, +) be an abelian group, and ${f:G\rightarrow H}$ . The second order Cauchy difference of f is $$C^{(2)}f(x,y,z)=f(xyz)-f(xy)-f(yz)-f(xz)+f(x)+f(y)+f(z).$$ The functional equation $$C^{(2)}f(x,y,z)=0$$ is studied. We present its general solution on free groups. Solutions on other selected groups are also given.     

On Sharing Values of Meromorphic Functions and Their Differences

For a meromorphic function f in the complex plane, we prove that if f is a finite order transcendental entire function which has a finite Borel exceptional value a, if ${f(z+\eta)\not\equiv f(z)}$ for some ${\eta\in \mathbb{C}}$ , and if f(z + η) ? f(z) and f(z) share the value a CM, then $$ a=0 \quad {\rm and} \quad \frac{f(z+\eta)-f(z)}{f(z)}=A, $$ where A is a nonzero constant. We also consider problems on sharing values of meromorphic functions and their differences when their orders are not an integer or infinite.     

关于代数体函数的展布关系及其应用

关于代数体函数的展布关系及其应用杨连中（山东大学数学系，济南２５０１００）国家青年自然科学基金资助课题．１９９１年５月１５日收到，１９９２年２月２１日收到修改稿．一、引言设ｆ（ｚ）于开平面亚纯，下级为有穷正数．若复数ａ是ｆ（ｚ）的一个亏值，则下述展布...     

Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate q-Bernstein polynomials for a function analytic in the polydisc ${D_{{R_1}}} \times {D_{{R_2}}} = \{ z \in C:\left| z \right| < {R_1}\} \times \{ z \in C:\left| z \right| < {R_1}\} $ for arbitrary fixed q > 1. We give quantitative Voronovskaja type estimates for the bivariate q-Bernstein polynomials for q > 1. In the univariate case the similar results were obtained by S.Ostrovska: q-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255, and S.G.Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.