Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.     

Explicit solutions of the equation –div z = f under the constraint ∥z ∥∞ = 1

We consider the eigenvalue problem for the 1-Laplace operator and the corresponding variational problem. Since the underlying functionals are not differentiable, the derivation of the Euler-Lagrange equation is highly nontrivial and involves methods of nonsmooth analysis. A remarkable fact is, that there are infinitely many functions f for which the equation –div z = f has to be satisfied. The purpose of the present work is the investigation of this equation. We show that in a square 2-dimensional domain for each right hand side f there are infinitely many vector fields z satisfying the Euler-Lagrange equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

论函数序列{f(λ_nΖ)}的完备性

<正> 在很多工作中(例如:A.O.Гепьфоонп[1],A,И.Мapkyшевия[2],И.И.Ибpaгиммов[3]И.Ф.Пoxин[4],R.P.Boas,Jr,[5],A,Ф.Пeoнтъев[6]研究了函数系{f(λ_nΖ)}是整函数,且 f~((n))(0)≠0,n=0,1,2,…)在圆|z|相似文献   

On Univalence of the Power Deformation z（f（z）/z）c

The authors mainly concern the set U _f of c ?? ? such that the power deformation $ z(\frac{{f(z)}} {z})^c $ is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a ₂ z ² + ?? in the unit disk. It is shown that U _f is a compact, polynomially convex subset of the complex plane ? unless f is the identity function. In particular, the interior of U _f is simply connected. This fact enables us to apply various versions of the ??-lemma for the holomorphic family $ z(\frac{{f(z)}} {z})^c $ of injections parametrized over the interior of U _f . The necessary or sufficient conditions for U _f to contain 0 or 1 as an interior point are also given.     

Transcendence of solutions of q-Painlev�� equation of type {A^{(1)}_{6}}

In this paper we prove the transcendence of the solutions of the q-Painlevé equation of type ${A_6^{(1)}}$ . The q-Painlevé equation of type ${A_6^{(1)}}$ is also called d-P _II or q-P _II and has a continuous limit to the Painlevé equation of type II. Its symmetry is (A ₁?+?A ₁)⁽¹⁾.     

Some Arithmetical Properties of the Generating Power Series for the Sequence {ζ(2k+1)}k=1 ∞

Let f_odd(z):= ∑^∞ _k=1ζ(2k + 1)z^2k be the power series with the values of the Riemann ζ function at odd integers as coefficients. This function can be analytically continued to a meromorphic function over C. We prove that 1 and the values of f_odd at rational points with relatively prime denominators are linearly independent over ―Q. Some arithmetical properties of the sequence {ζ(2k+1)} _k=1 ^∞ are deduced. This revised version was published online in June 2006 with corrections to the Cover Date.     

函数序列{f_(2j+1)(t)}与{g_(2j+1)(t)}的一致收敛性

由[1]中调配函数的性质3及5可知,当j为奇数时,f_(2j 1)(t)及g_(2j 1)(t)都是[0,1]上点(0,0)及点(1,0)之间的一段凹弧;当j为偶数时,为此二点间的一段凸弧. 下面仅就g_(2j 1)(t)来讨论.如上所述,可知g_(2j 1)(t)在[0,1]上是单调函数,且g_(2j 1)(0)与g_(2j 1)(1)异号,即     

On Divergence of Hermite—Fejér Interpolation to f(z)=z in the Complex Plane

We show that for a broad class of interpolatory matrices on [-1,1] the sequence of polynomials induced by Hermite—Fejér interpolation to f(z)=z diverges everywhere in the complex plane outside the interval of interpolation [-1,1] . This result is in striking contrast to the behavior of the Lagrange interpolating polynomials. June 15, 1998. Date accepted: January 26, 1999.     

Abschätzungen für\sum\limits_{n \leqslant N} {B_1 (n\alpha )}

Estimates for . Let and letB ₁(x)={x}–1/2. In this paper we shall give best possible estimates for . On the importance of this sum see the papers ofBehnke [2],Hardy andLittlewood [5],Hecke [6] andOstrowski [9].

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Meinem verehrten Lehrer, Herrn Professor E. Hlawka zum siebzigsten Geburtstag gewidmet     

The growth of solutions of f″+ e-zf′+ Q(z)f - 0 where the order (Q)＝ I

This paper investigates the growth of solutions of the equation f" + e-zf′ + Q(z)f＝ 0 wherethe order (Q)＝ 1. When Q(z) ＝ h(z)ebz, h(z) is nonzero polynomial, b ≠ -1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M.Ozawa, G. Gundersen and J. K. Langley.     

Hypergeometric τ-functions of the q-Painlevé system of type $E_{8}^{(1)}$

We present the τ-functions for the hypergeometric solutions to the q-Painlevé system of type E₈⁽¹⁾E_{8}^{(1)} in a determinant formula whose entries are given by Rahman’s q-hypergeometric integrals. By using the symmetry of the q-hypergeometric integral, we can construct 56 solutions and describe the action of W(E₇⁽¹⁾)W(E_{7}^{(1)}) on the solutions.     

微分方程f″ e~(az)f′ Q(z)f=F(z)的复振荡

该文研究了线性微分方程f″ eazf′ Q( z) f=F( z)的复振荡问题,其中Q( z)、F( z) ( 0 )是整函数,且σ( Q) =1 ,σ( F) < ∞,Q( z) =h( z) ebz,h( z)是多项式,b≠- 1是复常数,那么上述线性微分方程的所有解f( z)满足λ( f) =λ( f) =σ( f) =∞,　λ2 ( f) =λ2 ( f) =σ2 ( f) =1 .至多除去两个例外复数a及一个可能的有穷级例外解f0 ( z) .     

Evaluation of Series Involving the Product of the Tail of $${\zeta(k)}$$ and $${\zeta(k+1)}$$ζ(k+1)

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values

$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$

where \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.

     

Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.     

ON SET {N_k~-(x)}_k~n=0

Suppose that C is the complex plane and k is a non-negative integer.Define functions Nk-(x)=|x|k if k is even and Nk(x)=x|x|k-1 if k is odd.Some approximation properties of Nk-(x)'s is discussed and a new example of a Tchebycheff system is given out.     

The pluripolar hull of {w=e −1/z }

In this paper we show that the pluripolar hull ofE={(z, ω)∈C²:ω=e ^−1/z,z≠0} is equal toE. This implies thatE is plurithin at 0, which answers a question of Sadullaev. The result remains valid ife ^−1/z is replaced by certain other holomorphic functions with an essential singularity at 0.     

Convergence of simultaneous Hermite-Padé approximants to then-tuple ofq-hypergeometric series\{ {}_1\Phi _1 (_{(c,\gamma _j )}^{(1,1)} ;z)\} _{j = 1}^n

We investigate the convergence of simultaneous Hermite-Padé approximants to then-tuple of power series $$f_i (z) = \sum\limits_{k = 0}^\infty {C_k^{(i)} z^k ,} i = 1,2,...,n,$$ where $$C_0^{(i)} = 1;C_k^{(i)} = \prod\limits_{p = 0}^{k - 1} {\frac{1}{{(C - q^{\gamma i + p} )}},} k \ge 1.$$ HereC, q∈?, γ _i ∈?,i=1, 2,...,n. For |C|≠1, ifq=e^iθ, θ∈(0, 2π) and θ/2π is irrational, eachf _i (z),i=1,...,n, has a natural boundary on its circle of convergence. We show that “close-to-diagonal” and other sequences of Hermite-Padé approximants converge in capacity to (f ₁(z),..., f_n (z)) inside the common circle of convergence of eachf _i (z),i=1,...,n.