A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers
A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers
复制标题
模函数值的代数独立性准则及其在涉及斐波那契数和卢卡斯数的无限乘积中的应用
DOI：
10.1007/s40993-022-00328-7
复制
发表时间：
2022
期刊：
影响因子：
--
通讯作者：
and Yohei Tachiya
中科院分区：
文献类型：
--
作者：
Daniel Duverney;Carsten Elsner;Masanobu Kaneko;and Yohei Tachiya
The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \prod _{n=1}^{\infty }\left( 1+\frac{1}{F_n} \right) ,\quad \prod _{n=3}^{\infty }\left( 1-\frac{1}{F_n} \right) ,\quad \prod _{n=1}^{\infty }\left( 1+\frac{1}{L_n} \right) ,\quad \prod _{n=2}^{\infty }\left( 1-\frac{1}{L_n} \right) \end{aligned}$$\end{document}are algebraically independent over, whereandare the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series.
The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \prod _{n=1}^{\infty }\left( 1+\frac{1}{F_n} \right) ,\quad \prod _{n=3}^{\infty }\left( 1-\frac{1}{F_n} \right) ,\quad \prod _{n=1}^{\infty }\left( 1+\frac{1}{L_n} \right) ,\quad \prod _{n=2}^{\infty }\left( 1-\frac{1}{L_n} \right) \end{aligned}$$\end{document}are algebraically independent over, whereandare the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series.
登录
查看更多内容
影响因子：
0.7
作者：
Takuya Abe;Ikuro Shimizu;Yuya Murakami
通讯作者：
Yuya Murakami
某些无限乘积的代数独立性
DOI：
--
发表时间：
2009
期刊：
影响因子：
--
作者：
M.Amou;Yann Bugeaud;天羽雅昭
通讯作者：
天羽雅昭
雅可比theta级数的超越及相关结果
DOI：
10.1515/9783110809794.157
发表时间：
1998
期刊：
Proceedings of the Japan Academy, Series A, Mathematical Sciences
影响因子：
--
作者：
D. Duverney;K. Nishioka;K. Nishioka;I. Shiokawa
通讯作者：
I. Shiokawa
影响因子：
0.8
作者：
Y. Nesterenko
通讯作者：
Y. Nesterenko
马勒-马宁猜想的前奏
DOI：
--
发表时间：
1996
期刊：
影响因子：
--
作者：
Katia Barré;G. Diaz;F. Gramain;Georges Philibert
通讯作者：
Georges Philibert