A class of hypertranscendental functions
A class of hypertranscendental functions
复制标题
一类超超越函数
DOI：
10.1007/bf01836423
复制
发表时间：
1977
期刊：
影响因子：
--
通讯作者：
A. J. Poorten
中科院分区：
文献类型：
--
作者：
J. Loxton;A. J. Poorten
超超越函数是不满足任何代数微分方程的函数。这类函数的一些著名的例子是黎曼的ζ函数和代数数域的ζ函数。(见[4])。在这篇笔记中，我们研究了满足下面第8节描述的类型的泛函方程的函数的超超越。在第3节至第7节中，我们还得到了满足这类泛函方程的一组函数在有理函数域上代数独立的条件；事实上，这是我们在第8节证明超超越结果的主要步骤。最后，第9节包含了一些例子来说明前面的分析。在一系列以[3]告终的论文中，Mahler获得了一些关于满足这里所考虑的类型的泛函方程的函数值的超越性和代数独立性的非常一般的结果。论证的一部分是研究函数本身的超越性和代数独立性。（见b[3]， 549-556页）。我们的结果与马勒给出的结果具有相同的精神，但我们考虑的是更一般的一类泛函方程。我们也希望已经修复了[3]中的一些遗漏，因此[3]第554页的主要结果是不正确的。我们要感谢马勒教授让我们注意到他的论文b[3]，并感谢他在那里提出的思想。
A hypertranscendental function is one which does not satisfy any algebraic differential equation. Some well-known examples of such functions are the zeta function of Riemann and the zeta functions of algebraic number fields.(See [4]). In this note, we study the hypertranscendence of functions satisfying functional equations of the type described in Section 8 below. We obtain also, in Sections 3 to 7, conditions under which a set of functions satisfying functional equations of this type is algebraically independent over the field of rational functions; in fact this is the main step in the proof of the hypertranscendence result which we prove in Section 8. Finally, Section 9 contains some examples illustrative of the foregoing analysis.In a series of papers culminating in [3], Mahler has obtained some very general results on the transcendence and algebraic independence of the values of functions satisfying functional equations of the type considered here. One part of the argument is to investigate the transcendence and algebraic independence of the functions themselves.(See [3], pages 549-556). Our results are in the same spirit as those given by Mahler, but we consider a more general class of functional equations. We also hope to have repaired some omissions in [3], as a consequence of which the main result on page 554 of [3] is incorrect as stated. We would like to thank Professor Mahler for drawing our attention to his paper [3] and to acknowledge our debt to the ideas he develops there.