ON THE CHOICES OF ACCELERATING CONVERGENCE FACTORS FOR LIMIT PERIODIC CONTINUED FRACTION K(an/1)

There are many accelerating convergence factors (ACFs) for limit periodic continued fraction K(an/1)(an→a≠0). In this paper, some characteristics and comparative theorems are ob tained on ACFs. Two results are given for most frequently used ACFs.  

On the Convergence of Limit Periodic Continued Fractions K(an/1) where an→ - ¼. Part IV

Continued fractions K(a _n /b _n ) , where a _n , b _n ∈\smallbf C and a _n /b _n b _n-1 →-\frac 14 , may converge or diverge depending on how a _n /b _n b _n-1 approaches its limit. Due to equivalence transformations it suffices to study the special case where all b _n =1 . We shall prove that K(a _n /1) converges if a _n →-\frac 14 and there exists a set V\subseteq\smallbf C \cup{∈fty} with certain properties such that a _n /(1+V)\subseteq V for all n . We shall also summarize some other useful consequences of such value sets V . January 31, 2000. Date revised: July 28, 2000. Date accepted: August 16, 2000.  

一类极限循环连分式的加速收敛因子

本文获得了一类极限循环连分式的加速收敛因子，证明了它们具有良好的加速收敛性质.  

Continued fractions with multiple limits

For integers m2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem.We give a theorem on a class of Poincaré-type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity.We also generalize a curious q-continued fraction of Ramanujan's with three limits to a continued fraction with k distinct limit points, k2. The k limits are evaluated in terms of ratios of certain q-series.Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m2.