A Research on the Theory of Irrational Numbers for q-Functions

我们主要在q-函数的无理数理论方面得到了如下结果。我们估计了满足多项式系数线性q-差分方程的函数的某些特殊值所生成的向量空间的维数。首席研究员与Matala-aho教授（欧卢大学）一起提交了一篇关于该结果的论文。我们证明了一个完整的模拟Shidlovskii引理，无论是帕德型近似的第一和第二类，一个系统的q-差分方程的系数从有理函数。此外，我们还应用这一结果得到了满足上述系统的解析函数的特殊值的线性无关性。首席研究员正在与Vaananen教授（欧卢大学，来自国外的研究员）和Matala-aho教授一起准备一份关于该结果的论文。本文用Duverney-Nishioka关于Mahler函数的归纳方法证明了Galochkin关于函数线性无关性的一个判别准则，并给出了一个对Duverney-Nishioka的结果进行定量加细的方法.研究负责人与Vaananen教授一起准备研究结果的论文，在有关基金的支持下，研究负责人访问了白俄罗斯（2003年8月）和芬兰（2003年9月; 2005年9 - 10月），并邀请了Kohnen教授和Lasjaunias教授（2004年10月），他们对本研究非常重要。

We have obtained the following results mainly on the theory of irrational numbers for q-functions.1. We estimated from above the dimension of the vector space generated by certain special values of functions satisfying linear q-difference equations with polynomial coefficients. The head investigator submitted a paper on the result with Prof.Matala-aho (University of Oulu).2. We proved a complete analogue of Shidlovskii's lemma, both on Pade-type approximations of the first and the second kinds, for a system of q-difference equations with coefficients from rational functions. Moreover, we apply this result to get linear independence of special values of analytic functions satisfying the above system. The head investigator is preparing a paper on the result with Prof.Vaananen (University of Oulu, a investigator from abroad) and Prof.Matala-aho.3. We convined a criterion on linear independence of functions given by Galochkin with an inductive method in Mahler functions given by Duverney-Nishioka, and created a method to obtaining a quantitative refinement of a resul of Duverney-Nishioka. The head investigator is preparing a paper on the result with Prof.Vaananen.By the support of the grant concerned, the head investigator visited Belarus (August, 2003) and Finland (September, 2003 ; September-October, 2005), and invited Prof.Kohnen and Prof.Lasjaunias (October, 2004), which were very important for the present research.