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
DOI:
https://doi.org/10.1090/S0002-9947-1972-0315107-4
MathSciNet review:
0315107
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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.
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Additional Information
Keywords:
Continued fraction,
convergence region,
admissable sequence,
linear fractional transformation
Article copyright:
© Copyright 1972
American Mathematical Society