A Research on Arithmetical Properties of Special Values of q-Functions

研究了某些q-函数特殊值的算术性质，得到了如下结果.本文讨论了一类q-超几何级数，确定了其值为无理数的有理数。首席研究员与Masanori Katsurada教授（庆应义塾大学，研究人员之一）和凯乔·瓦纳宁教授（欧卢大学）。1999年11月，在京都举行的解析数论会议上，他与桂田教授一起就这些结果发表了演讲。我们处理一个解（f_1，...，f_m），并证明了f_1，.，f_2，.，f_3，...，f_m是一个非零有理数向量。这一结果部分量化了J. P.Bezivin（Mandarepta Math.，1988年）。首席研究员与Vaananen教授一起就此结果撰写了一篇论文。3.证明了满足Poincare型函数方程的q-函数在E-函数中的一个类似的Shidlovskii引理。Tchakaloff函数和q-指数函数的情况被认为是特殊情况。研究负责人与Vaananen教授一起就这一结果写了一篇论文。在有关基金的支持下，研究负责人访问了Alain Lasjaunias教授（波尔多大学）。两次（1999年8月; 2000年8月）和Vaananen教授两次（1999年9 - 10月; 2000年9 - 10月）。特别是，后来的访问对目前的研究非常重要。

We have studied arithmetical properties of special values of certain q-functions and obtained the following results.1. We treated a class of q-hypergeometric series and determined the rational numbers for which the values of the series are irrational numbers. The head investigator published two papers on these results with Prof.Masanori Katsurada (Keio Univ., one of the investigators) and Prof.Keijo Vaananen (Univ.of Oulu). He also gave a talk on these results with Prof.Katsurada at the conference on analytic number theory held in Kyoto (November, 1999).2. We treated a solution (f_1, ..., f_m) of a system of m functional equations of Poincare type in several variables, and proved linear independence of the values of f_1, ..., f_m at a vector of nonzero rational numbers in quantitative nature. This result partly quantifies a result of Prof.J.-P.Bezivin (Manuscripta Math., 1988). The head investigator wrote a paper on this result with Prof. Vaananen.3. We proved an analogue of Shidlovskii's lemma in E-functions for q-functions satisfying certain functional equations of Poincare type. The case for Tchakaloff functions and that for q-exponential functions were considered as paticular cases. The head investigator wrote a paper on this result with Prof.Vaananen.By the support of the grant concerned, the head investigater visited Prof.Alain Lasjaunias (Bordeaux Univ.) twice (August, 1999 ; August, 2000) and Prof.Vaananen twice (September-October, 1999 ; September-October, 2000). In particular, the later visit was very important for the present research.