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首页> 外文会议>International Symposium on Symbolic and Algebraic Computation Aug 3-6, 2003 Philadelphia, Pennsylvania, USA >Power Series, Bieberbach Conjecture and the de Branges and Weinstein Functions
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Power Series, Bieberbach Conjecture and the de Branges and Weinstein Functions

机译：幂级数，Bieberbach猜想以及de Branges和Weinstein函数

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摘要

It is well-known that de Branges' original proof of the Bieberbach and Milin conjectures on the coefficients a_n of univa-lent functions f(z) = Σ_(κ=1)~∞ a_nz~n of the unit disk as well as Weinstein's later proof deal with the same special function system that de Branges had introduced in his work. These hypergeometric polynomials had been already studied by Askey and Gasper who had realized their positiveness. This fact was the essential tool in de Branges' proof. In this article, we show that many identities, e.g. the representation of their generating function w.r.t. n, for these polynomials, which are intimately related to the Koebe function K(z) = Σ_(κ=1)~∞ nz~n and therefore to univalent functions, can be automatically detected from power series computations by a method developed by the author and accessible in several computer algebra systems. In other words, in this paper a new and interesting application of the FPS (Formal Power Series) algorithm is given. As working engine we use a Maple implementation by Dominik Gruntz and the author. In particular, the hypergeometric representation of both the de Branges and the Weinstein functions are determined by successive power series computations from their generating functions. The new idea behind this algorithm is the observation that hypergeometric function coefficients of double series can be automatically detected by an iteration of the FPS procedure. In a final section we show how algebraic computation enables the fast verification of Askey-Gasper's positivity results for specific (not too large) n using Sturm sequences or similar approaches.
机译：众所周知，de Branges对比伯巴赫和米林猜想的原始证明是关于单位圆盘的单项函数f（z）=Σ_（κ= 1）〜∞a_nz〜n的系数a_n以及韦恩斯坦的后来的证明处理了de Branges在他的工作中引入的相同的特殊功能系统。这些超几何多项式已经被Askey和Gasper研究，他们意识到了它们的积极性。这是de Branges证明中必不可少的工具。在本文中，我们显示了许多身份，例如它们的生成函数的表示n，对于这些与Koebe函数K（z）=Σ_（κ= 1）〜∞nz〜n密切相关的多项式，因此与单价函数密切相关的多项式，可以通过以下方法从幂级数计算中自动检测出来：作者，可以在多个计算机代数系统中访问。换句话说，在本文中，给出了FPS（形式幂级数）算法的新的有趣应用。作为工作引擎，我们使用Dominik Gruntz及其作者的Maple实现。尤其是，de Branges和Weinstein函数的超几何表示都由它们的生成函数通过连续幂级数计算确定。该算法背后的新思想是可以通过FPS程序的迭代自动检测双序列的超几何函数系数。在最后一部分中，我们将展示代数计算如何使用Sturm序列或类似方法快速验证特定（不是太大）n的Askey-Gasper阳性结果。

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