Tasoev's continued fraction and its applications

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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