Tasoev's continued fraction and its applications

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被很好地定义。应用Hurwitz连分式有三种类型的例子，即tanh型、tan型和e型，以更一般的方式构造了类似类型的Tasoev连分式，此外，还发现了与这些一般类型对应的Hurwitz连分式。2.Tasoev连分式与其他连分式的关系通过发现Tasoev连分式和Hurwitz连分式在部分商序列分别是几何的或算术的观点下有很好的比较，一旦在两个连分式中得到了新的结果，相应的新结果也在另一个连分式中得到。这种结构在未来也将得到应用。Rogeres-Ramanujan连分式，Gen…之一分子不总是1的更多的外在连分数，在某些情况下可以转化为简单的连分数，这就成为Tasoev连分数。3.建立了Tasoev连分式的有理逼近的求值方法，使产生Tasoev连分式的无理数用有理数很好地逼近。对于一些具体的Tasoev连分式，这种评价的精确值是能够实际给出的。4.连分式的新应用发展了一种新的非齐次连分式展开算法，用于求解非齐次丢番图问题。定义了由连分式的每n个收敛点组成的跳跃收敛点，并发现了它们的各种有趣的性质。Zaremba猜想在有限连分式中仍然是一个悬而未决的问题，如果有理数的分母是偶数次幂的话，这个猜想就部分解决了。较少

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