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

Jones, William B.; Snell, R. I.

doi:10.1090/S0002-9947-1972-0315107-4

Sequences of convergence regions for continued fractions $K(a_{n}/1)$
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by William B. Jones and R. I. Snell PDF

Trans. Amer. Math. Soc. 170 (1972), 483-497 Request permission

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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