Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Sequences of convergence regions for continued fractions $K(a_{n}/1)$

Authors: William B. Jones and R. I. Snell
Journal: Trans. Amer. Math. Soc. 170 (1972), 483-497
MSC: Primary 30A22
MathSciNet review: 0315107
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions are given for convergence of continued fractions $K({a_n}/1)$ such that ${a_n} \in {E_n},n \geqslant 1$, where $\{ {E_n}\}$ is a sequence of element regions in the complex plane. The method employed makes essential use of a nested sequence of circular disks (inclusion regions), such that the $n$th disk contains the $n$th approximant of the continued fraction. This sequence can either shrink to a point, the limit point case, or to a disk, the limit circle case. Sufficient conditions are determined for convergence of the continued fraction in the limit circle case and these conditions are incorporated in the element regions ${E_n}$. The results provide new criteria for a sequence $\{ {E_n}\}$ with unbounded regions to be an admissible sequence. They also yield generalizations of certain twin-convergence regions.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A22

Retrieve articles in all journals with MSC: 30A22

Additional Information

Keywords: Continued fraction, convergence region, admissable sequence, linear fractional transformation
Article copyright: © Copyright 1972 American Mathematical Society