Tasoev's continued fraction and its applications

塔索耶夫连分数及其应用

• 批准号: 15540021
• 负责人: KOMATSU Takao
• 金额: $ 1.02万
• 依托单位: Hirosaki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540021/
• 关键词: Continued fractions; Tasoev continued fractions; Hurwitz continued fractions; Diophantine approximations; Rational approximations; Diophantine problems; Rogers-Ramanujan; Zaremba's conjecture; Rogers-Ramanujan continued fractions; Tasoev's contin

1.Definition and discovery of Tasoev continued fractionsSome basic types of examples of the Tasoev continued fractions were discovered and were well-defined. Applying the fact that Hurwitz continued fractions had three types of examples, namely, tanh-type, tan-type and e-type, the similar types of Tasoev continued fractions were constructed in a more general way, then in addition, the Hurwitz continued fractions corresponding to these general types were also discovered.2.Relation between Tasoev continued fraction and other continued fractionsBy finding that the Tasoev continued fraction and the Hurwitz continued fraction have a good comparison under the viewpoint that the sequence of partial quotients is geometric or arithmetic, respectively, once new results were obtained in either continued fraction, the corresponding new results were also obtained in another continued fraction. This structure will be applied in the future too. The Rogeres-Ramanujan continued fraction, one of the gen … More eral continued fractions whose numerators are not always 1,can be converted to the simple continued fraction in some cases, which becomes a Tasoev continued fraction. Some concrete examples were also given.3.Rational approximation of Tasoev continued fractionThe way of evaluation so that the irrational number yielding Tasoev continued fraction is well-approximated by the rational number, was established. The exact values of such evaluations for some concrete Tasoev continued fractions were able to be given actually.4.New applications of continued fractionA new inhomogeneous continued fraction expansion algorithm was developed in the problem to obtain a value in inhomogeneous Diophantine problems. Leaping convergents, which are composed from every n-th convergents of the continued fraction, were defined, and their various interesting characters were found. The Zaremba's conjecture, which is a still open problem in finite continued fractions, were partially solved if the denominators of rational numbers had even powers. Less

1.Tasoev 连分式的定义和发现Tasoev 连分式的一些基本类型的例子被发现并得到了很好的定义。利用Hurwitz连分数具有tanh型、tan型和e型三种类型的事实，以更通用的方式构造了Tasoev连分数的类似类型，此外还发现了这些通用类型对应的Hurwitz连分数。 2.Tasoev连分数与其他连分数的关系通过发现Tasoev连分数 分式与Hurwitz连分式有很好的可比性，从部分商的序列分别是几何或算术的角度来看，一旦在任何一个连分式中得到新的结果，在另一个连分式中也得到相应的新结果。这种结构在未来也会得到应用。 Rogeres-Ramanujan 连分数是分子不总是 1 的通用连分数之一，在某些情况下可以转换为简单连分数，即成为 Tasoev 连分数。并给出了具体实例。 3.Tasoev连分式的有理逼近 建立了用有理数很好地逼近产生Tasoev连分式的无理数的求值方法。对于一些具体的Tasoev连分式，实际能够给出这种评估的准确值。4.连分式的新应用 针对该问题，提出了一种新的非齐次连分式展开算法，以求得非齐次丢番图问题的值。定义了由连分数的每n个收敛组成的跳跃收敛，并发现了它们各种有趣的性质。扎伦巴猜想是有限连分数中的一个仍然悬而未决的问题，如果有理数的分母具有偶次幂，则可以部分解决。较少的

项目成果

Rational approximations to Tasoev continued fractions, II

Tasoev 连分数的有理逼近，II

• 发表时间: 2005
• 期刊: Lietuvos Matematikos Rinkinys 45-1
• 作者: C.H.Lam;H.Yamada;Takao Komatsu;Takao Komatsu
• 通讯作者: Takao Komatsu

連分数から実数へ

从连分数到实数

• 发表时间: 2004
• 期刊: 数理解析研究所講究録 1384
• 作者: Takao Komatsu

Takao Komatsu: "The interval associated with a Fibonacci number"Fibonacci Quarterly. 41・1. 3-6 (2003)

小松隆雄：“与斐波那契数相关的区间”斐波那契季刊 41・1（2003）。

Some recognizable forms of simple continued fractions

简单连分数的一些可识别形式

• 发表时间: 2005
• 期刊: International Journal of Mathematical Sciences 4・1
• 作者: K.Hiraga;H.Saito;Hiraku Nakajima;Yo Matsubara;Takao Komatsu
• 通讯作者: Takao Komatsu

On a Zaremba's conjecture for powers

关于扎伦巴幂猜想

• 发表时间: 2005
• 期刊: Sarajevo Journal of Mathematics 1(13)・1
• 作者: M.Amou;Y.Bugeaud;T.Takebe;K.Takasaki;K.Takasaki;K.Takasaki;T.Takebe;Takao Komatsu
• 通讯作者: Takao Komatsu