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

我们研究了某些q函数的特殊值的算术性质，得到了以下结果：我们处理了一类q-超几何级数，并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keijo Vaananen教授（奥卢大学）就这些结果发表了两篇论文。他还在1999年11月于京都举行的解析数论会议上与katsurada教授就这些结果作了报告。我们处理一个溶液(f_1，…， f_m)，并证明了f_1，…的值的线性无关性。， f_m在非零有理数向量上的定量性质。这个结果在一定程度上量化了j.p教授的结果。贝兹文（数学手稿）, 1988)。首席研究员与vaananen教授就这一结果写了一篇论文。对于满足某些庞加莱型泛函方程的q函数，证明了e函数中Shidlovskii引理的一个类似。恰卡洛夫函数的情况和q指数函数的情况被视为特殊情况。首席研究员与vaananen教授就这一结果撰写了一篇论文。在有关经费的支持下，首席研究员两次（1999年8月、2000年8月）拜访了alain Lasjaunias教授（波尔多大学），两次（1999年9月至10月、2000年9月至10月）拜访了vaananen教授。特别是后来的访问对现在的研究非常重要。

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.