关于连分数的构造

Observations on continued fractions

This is a translation of Euler's Latin paper "De fractionibus continuis observationes" into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the Riccati-Differential equation by means of continued fractions and finds many other interesting formulas and results (e.g, the continued fraction for the quotient of two hypergeometric series usually attributed to Gauß)

由素数构造的连分数

Continued fractions constructed from prime numbers

给出了由a）素数，b）孪生素数，c）广义$d$ -孪生，d）$m^2+n^4$形式的素数，e）$m^2+1$形式的素数，f）Mersenne素数和g）素素数构成的简单连分数的50位数的值.所有这些连分数都属于Khinchin和Levy定理例外的零测度集。我们认为所有这些连分数都是超越数。其次，我们提出了一个猜想，它指出了从某些连分数的超越性推导另一些连分数的超越性的方法。

如何证明Ramanujan的$q$ -连分数

How to prove Ramanujan's $q$-continued fractions

利用欧拉的方法，利用欧几里德算法展开一个幂级数为连分数，我们展示了如何推导拉马努金的$q$ -连分数在一个系统的方式。

拉马努金初等连分数的初等处理

An elementary treatment of Ramanujan's elementary continued fractions

本文研究拉马努金第二笔记本第12章中的初等连分数，这些连分数是其超几何与$q$-超几何连分数的前身，并给出初等处理。

函数sec（x）+ tan（x）的连分数展开

Continued fraction expansion for function sec(x) + tan(x)

给出了函数sec（x）+ tan（x）的连分数展开式的初等推导.

Some Continued Fractions for $π$ AND $G$

We present here two classes of infinite series and the associated continued fractions involving $π$ and Catalan's constant $G$ based on the work of Euler and Ramanujan. A few sundry continued fractions are also given.