Sequences of convergence regions for continued fractions

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 such that , where 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 th disk contains the 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 . The results provide new criteria for a sequence with unbounded regions to be an admissible sequence. They also yield generalizations of certain twin-convergence regions.

**[1]**T. L. Hayden,*A convergence problem for continued fractions*, Proc. Amer. Math. Soc. 14 (1963), 546-552. MR**27**#3961. MR**0154001 (27:3961)****[2]**K. L. Hillam and W. J. Thron,*A general convergence criterion for continued fractions*, Proc. Amer. Math. Soc. 16 (1965), 1256-1262. MR**33**#1613. MR**0193393 (33:1613)****[3]**Williams B. Jones and R. I. Snell,*Truncation error bounds for continued fractions*, SIAM J. Numer. Anal. 6 (1969), 210-221. MR**40**#1000. MR**0247737 (40:1000)****[4]**William B. Jones and W. J. Thron,*Convergence of continued fractions*, Canad. J. Math.**20**(1968), 1037-1055. MR**37**#6446. MR**0230888 (37:6446)****[5]**-,*Twin-convergence regions for continued fractions*, Trans. Amer. Math. Soc. 150 (1970), 93-119. MR**41**#8640. MR**0264043 (41:8640)****[6]**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**11**, 244. MR**0032034 (11:244b)****[7]**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**26**#6887. MR**0116092 (22:6887)****[8]**W. J. Thron,*Convergence of sequences of linear fractional transformations and of continued fractions*, J. Indian Math. Soc. 27 (1963), 103-127. MR**32**#1331. MR**0183855 (32:1331)****[9]**H. S. Wall,*Analytic theory of continued fractions*, Van Nostrand, Princeton, N. J., 1948. MR**10**, 32. MR**0025596 (10:32d)**

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

Retrieve articles in all journals with MSC: 30A22

Additional Information

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