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 $K(a_{n}/1)$

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
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.

T. L. Hayden, A convergence problem for continued fractions, Proc. Amer. Math. Soc. 14 (1963), 546–552. MR 154001, DOI https://doi.org/10.1090/S0002-9939-1963-0154001-9
K. L. Hillam and W. J. Thron, A general convergence criterion for continued fractions $K(a_{n}/b_{n})$, Proc. Amer. Math. Soc. 16 (1965), 1256–1262. MR 193393, DOI https://doi.org/10.1090/S0002-9939-1965-0193393-3
William B. Jones and R. I. Snell, Truncation error bounds for continued fractions, SIAM J. Numer. Anal. 6 (1969), 210–221. MR 247737, DOI https://doi.org/10.1137/0706019
William B. Jones and W. J. Thron, Convergence of continued fractions, Canadian J. Math. 20 (1968), 1037–1055. MR 230888, DOI https://doi.org/10.4153/CJM-1968-101-3
William B. Jones and W. J. Thron, Twin-convergence regions for continued fractions $K(a_{n}/1)$, Trans. Amer. Math. Soc. 150 (1970), 93–119. MR 264043, DOI https://doi.org/10.1090/S0002-9947-1970-0264043-9
R. E. Lane and H. S. Wall, Continued fractions with absolutely convergent even and odd parts, Trans. Amer. Math. Soc. 67 (1949), 368–380. MR 32034, DOI https://doi.org/10.1090/S0002-9947-1949-0032034-8
L. J. Lange and W. J. Thron, A two-parameter family of best twin convergence regions for continued fractions, Math. Z. 73 (1960), 295–311. MR 116092, DOI https://doi.org/10.1007/BF01215312
W. J. Thron, Convergence of sequences of linear fractional transformations and of continued fractions, J. Indian Math. Soc. (N.S.) 27 (1963), 103–127 (1964). MR 183855
H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948. MR 0025596