形式级数域上Oppenheim连分数展式的一些算术性质

Some Arithmetic Properties of Oppenheim Continued Fraction Expansion over the Field of Formal Series

摘要: 本人出于对形式级数域上Engel连分数展式的算术性质和度量性质的学习和研究，而Engel连分数展式作为Oppenheim连分数展式的一个特例，考虑从特殊到一般的方法，在本文中，我们主要研究形式级数域上Oppenheim连分数展式的算术性质。主要结果包括该展式的有限性、收敛性和唯一性。本文的结论在Engel连分数展式，Sylvester连分数展式，正规连分数展式等这些特例中也成立，我们的结果更具有一般性和优越性，这有利于我们对形式级数域上的连分数展式有进一步的了解。

Abstract: I am interested in learning and researching the arithmetic and metric properties of Engel contin-ued fraction expansions over formal series fields, the Engel continued fraction expansion is a special case of Oppenheim fraction expansion, consider the method of moving from specific to general, in this article, we mainly study the arithmetic properties of Oppenheim continued fraction expansions over formal series fields. The main results include the finiteness, convergence and uniqueness of the expansion, the conclusion of this paper also holds in the special cases of Engel continued fraction expansion, Sylvester continued fraction expansion and normal continued fraction expansion, our results are more general and superior, which is beneficial to our understanding of continued fraction expansions over formal series fields.