A Research on Transcendence and Algebraic Independence of Special Values of Automorphic Functions

我们研究了某些Q-函数的特殊值，得到了如下结果。设K是有限次代数数域，q是K中的元素，|q|upsilon&gt；1.L是正整数，q(Z)和R(Z)多项式的系数都满足deg q&lt；小于等于&gt；L，q(0)&lt；Double Plus&gt；0.在这些记号下，我们考虑形式为Q(Z)f(Qz)=z^lf(Z)+R(Z)的泛函方程，它有唯一解f(Z)在整个K&lt；upsilon&gt；上亚纯，K关于upsilon的完备化。在鹿儿岛大学教授Masanori Katsurada的帮助下，我们证明了下面的定理。和Keijo V_n_rien教授(大学Oulu)。(设α是K的非零元素，它不是f(Z)的极点。则f(α)不属于K。此外，存在一个有效可计算的正常数u，使得f(α)相对于上界的无理性度量由u自上而下追逐。特别地，f(α)不是Liouville数。)该定理对某些q-超几何级数有重要的应用，推广了Stihl(Math.在有关基金的支持下，我们邀请了Alain Lasjaunias教授(波尔多大学)。去群马大学。1998年10月，并询问了他对上述结果的看法，并与他讨论了进一步研究扩展的可能性。关于目前的研究，我们在1998年11月在东京举行的法日超越数论会议上作了一次报告，目前正在与片田教授和V_n_nen教授共同准备一篇论文。

We have studied special values of certain q-functions and obtained the following result. Let K be an algebraic number field of finite degree, upsilon a place of K, and q an element of K with |q|upsilon> 1. Let l be a positive integer, and Q(z) and R(z) polynomials with coefficients from K satisfying deg Q <less than or equal> l, Q(0) <double plus> 0. Under these notations, we consider a functional equation of the form Q(z)f(qz) = z^lf(z) + R(z), which has the unique solution f(z) meromorphic on the whole K_<upsilon>, the completion of K with respect to upsilon. Then we proved the following theorem with the aid of Prof. Masanori Katsurada (Kagoshima Univ.) and Prof. Keijo V__n_rien (Univ. of Oulu). (Let alpha be a nonzero element of K which is not a pole of f(z). Then f(alpha) does not belong to K.Moreover, there exists an effectively computable positive constant mu such that the irrationality measure of f(alpha) relative to upsilon is hounded from above by mu. In particular, f(alpha) is not a Liouville number.) This theorem has an important application to certain q-hypergeometric series, by which we can generalize a result of Stihl (Math. Ann., 1984).By the support of the grant concerned, we invited Prof. Alain Lasjaunias (Bordeaux Univ.) to Gunma Univ. on October, 1998, and asked his opinion about the above mentioned result and also discussed with him about a possibility of further researches extending it. Concerning the present research, we gave a talk at the conference between France and Japan on the theory of transcendental numbers held at Tokyo on November, 1998, and are now preparing a joint paper with Prof. Katsurada and Prof. V__n_nen.