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

我们研究了某些Q-函数特殊值的算术性质，得到了如下结果。研究了一类q-超几何级数，确定了其值为无理数的有理数。首席调查员与Masanori Katsurada教授(庆应大学，调查人员之一)和Keijo Vaananen教授(Oulu大学)就这些结果发表了两篇论文。他还在京都举行的解析数论会议(1999年11月)上与Katsurada教授就这些结果进行了讨论。本文讨论了m个Poincare型多变量泛函方程组的解(f_1，…，f_m)，证明了f_1，…，f_m在一个非零有理数向量上的数值在数量上是线性无关的。这一结果部分量化了J.-P.Bezivin教授(Manuscripta Math.，1988)的结果。首席研究员与Vaananenen教授就这一结果撰写了一篇论文。对于满足某些Poincare型函数方程的Q-函数，我们证明了E-函数中Shidlovskii引理的类似性。Tchakaloff函数和q指数函数的情形被视为特殊情况。首席调查员与Vaananen教授就这一结果撰写了一篇论文。在有关赠款的支持下，首席调查员访问了Alain Lasjaunias教授(波尔多大学)。两次(1999年8月；2000年8月)和两次Vaananen教授(1999年9月至10月；2000年9月至10月)。尤其是后一次访问，对于本研究来说是非常重要的。

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.