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Transactions of the American Mathematical Society

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

 

 

Sequences of convergence regions for continued fractions $ K(a\sb{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: 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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DOI: https://doi.org/10.1090/S0002-9947-1972-0315107-4
Keywords: Continued fraction, convergence region, admissable sequence, linear fractional transformation
Article copyright: © Copyright 1972 American Mathematical Society