Tasoev's continued fraction and its applications

1. Tasoev连分式的定义与发现发现了Tasoev连分式的一些基本类型的例子，并给出了它们的定义。应用Hurwitz连分数有三种类型的例子，即tanh型、tan型和e型，以更一般的方式构造了相似类型的Tasoev连分数，此外，2. Tasoev连分式与其它连分式的关系通过发现Tasoev连分式和Hurwitz连分式都是一种特殊的连分式，因此，本文提出了一种新的连分式，它是一种新的连分式在部分连续项序列分别是几何或算术的观点下，连分式具有很好的可比性，一旦在任一连分式中得到新结果，则在另一连分式中也得到相应的新结果。今后也将采用这种结构。Rogeres-Ramanujan连分式是广义连分式之一， 关于我们 分子不一定为1的连分式在某些情况下可以转化为简单连分式，从而成为Tasoev连分式。3. Tasoev连分式的有理逼近建立了使生成Tasoev连分式的无理数能很好地逼近有理数的求值方法。实际上能够给出某些具体Tasoev连分数的此类评估的精确值。4.连分数的新应用在该问题中开发了一种新的非齐次连分数展开算法，以获得非齐次丢番图问题的值。定义了由连分数的每n个连续点组成的跳跃连续点，并发现了它们的各种有趣的性质，在有理数的幂乘子为偶数的情况下，部分地解决了有限连分数中一个尚未解决的Zaremba猜想.少

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