A Research on Linear Independence of Special Values of q-Functions

q函数特殊值的线性无关性研究

• 批准号: 13640007
• 负责人: AMOU Masaaki
• 金额: $ 1.92万
• 依托单位: Gunma University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640007/
• 关键词: q- Function; q-Hypergeometric Sereis; Irrationality Measure; Linear Independence; Rationality Criterion; Theta Series; Linear Recurrence Sequence

We have studied linear independence of special values of certain q-functions and obtained the following results.1. We proved linear independence of the values of functions satisfying linear second order q-difference equations. The head investigator published a paper on the result (Proceedings of the China-Japan Seminar, in press) with Prof. T. Matala-aho (University of Oulu).2. We considered the values of power series introduced by Bezivin and proved linear independence of them qualitatively, which generalized earlier results obtained by Bezivin. The head investigator published a paper on the result (The Ramanujan J., in press) with Prof. K. Vaananen (University of Oulu, a investigator from abroad).3. We considered the values of generalized Tschakaloff series and obtained quantitative irrationality and linear independence results. The head investigator published a paper on the result (J. Number Theory, in press) with Prof. M. Katsurada (Keio University, one of the investigators).4. We proved linear independence of the values of theta series. The head investigator submitted a paper on the result with Prof. Vaananen.By the support of the grant concerned, the head investigator visited Prof. Vaananen and Prof. Matala-aho twice (August-September, 2001 ; August-September, 2002), which was very important for the present research.

我们研究了某些q函数的特殊值的线性无关性，得到了以下结果：证明了满足线性二阶q差分方程的函数值的线性无关性。首席研究员与T. Matala-aho教授（奥卢大学）共同发表了关于该结果的论文（《中日研讨会论文集》，已出版）。我们考虑Bezivin引入的幂级数的值，并定性地证明了它们的线性无关性，推广了Bezivin先前得到的结果。首席研究员与K. Vaananen教授（奥卢大学，国外研究员）就这一结果发表了一篇论文（The Ramanujan J.，出版）。我们考虑广义Tschakaloff序列的值，得到了数量上的不合理性和线性无关的结果。首席研究人员与M. Katsurada教授（庆应义塾大学，研究人员之一）一起发表了关于该结果的论文（J. Number Theory，已出版）。我们证明了级数值的线性无关性。首席研究员和瓦纳宁教授一起提交了一篇关于结果的论文。在相关基金的支持下，首席研究员两次拜访Vaananen教授和Matala-aho教授（2001年8月- 9月；2002年8月- 9月），这对本研究非常重要。

项目成果

M. Amou, M. Katsurada, and K. Vaananen: "On the values of certain q-hypergeometric series II"Analytic Number Theory ; the joint Proceedings of the China-Japan Number Theory Conference. 17-25 (2002)

M. Amou、M. Katsurada 和 K. Vaananen：“论某些 q 超几何级数 II 的值”解析数论；

M. Amou and M. Katsurada: "Irrationality results for values of generalized Tschakaloff series II"Aequationes Math. in press.

M. Amou 和 M. Katsurada：“广义 Tschakaloff 级数 II 值的非理性结果”数学方程。

M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series II"Analytic Number Theory ; the joint Proceedings of the China-Japan Number Theory Conference. 17-25 (2002)

M.Amou、M.Katsurada、K.Vaananen：“关于某些 q-超几何级数 II 的值”解析数论；

M.Amou, T.Matala-aho: "Arithmetic properties of solutions of linear second order q-difference equations"Number Theoretic Methods-Future Trends ; the Proceedings of the China-Japan Number Theory Conference. (印刷中).

M.Amou，T.Matala-aho：“线性二阶q-差分方程解的算术性质”数论方法-未来趋势；中日数论会议论文集（出版中）。

M.Amou, T.Matala-Aho: "Arithmetical properties of solutions of linear second order q-difference equations"Number Theoretic Methods-Future Trends ; the Proceedings of the China-Japan Number Theory Conference. (印刷中).

M.Amou，T.Matala-Aho：“线性二阶q-差分方程解的算术性质”数论方法-未来趋势；中日数论会议论文集（出版中）。