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

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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.