Transactions of the American Mathematical Society

Author(s): Lisa Lorentzen
Journal: Trans. Amer. Math. Soc. 360 (2008), 4287-4304.
MSC (2000): Primary 40A15
Posted: March 12, 2008
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Abstract: We prove that if the continued fraction $K(a_{n}/1)$ has circular twin value sets $\langle V_{0},V_{1}\rangle$ , then $K(a_{n}/1)$ converges except in some very special cases. The results generalize previous work by Jones and Thron.

1.

Jacobsen, Lisa, Some periodic sequences of circular convergence regions, Lecture Notes in Math. No 932, Springer-Verlag (1982), 87-98. MR 690456 (84c:40003)

2.

Jacobsen, Lisa, General convergence of continued fractions, Trans. Amer. Math. Soc. 294(2) (1986), 477-485. MR 825716 (87j:40004)

3.

Jacobsen, Lisa and Thron, W.J., Oval Convergence Regions and Circular Limit Regions for Continued Fractions $K(a_{n}/1)$ , Lecture Notes in Math., Springer-Verlag 1199 (1986), 90-126. MR 870246 (88c:40004)

4.

Jacobsen, Lisa and Thron, W.J., Limiting structures for sequences of linear fractional transformations, Proc. Amer. Math. Soc. 99 (1987), 141-146. MR 866444 (88d:40010)

5.

Jones, William B. and Thron, W.J., Convergence of continued fractions, Can. J. Math. 20 (1968), 1037-1055. MR 0230888 (37:6446)

6.

Jones, William B. and Thron, W.J., Twin-convergence regions for continued fractions $K(a_{n}/1)$ , Trans. Amer. Math. Soc. 150 (1970), 93-119. MR 0264043 (41:8640)

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Jones, William B. and Thron, W.J., Continued Fractions. Analytic Theory and Applications. Encyclopedia of Mathematics and Its Applications 11, Addison-Wesley Publishing Company, 1980. MR 0595864 (82c:30001)

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Lane, R.E., The Convergence and Values of Periodic Continued Fractions, Bull. Amer. Math. Soc. 51 (1945), 246-250. MR 0011748 (6:211b)

9.

Lorentzen, Lisa, Convergence criteria for continued fractions $K(a_{n}/1)$ based on value sets, Contemporary Mathematics 236 (1999), 205-255. MR 1665372 (2000i:30006)

10.

Lorentzen, Lisa, Möbius transformations mapping the unit disk into itself, The Ramanujan J. Math. 13 (2007), 253-263. MR 2281165

11.

Lorentzen, Lisa and Ruscheweyh, St., Simple convergence sets for continued fractions $K(a_{n}/1)$ , Math. Anal. and Appl. 179(2) (1993), 349-370. MR 1249825 (95b:40001)

12.

Lorentzen, Lisa and Waadeland, Haakon, Continued Fractions with Applications, Studies in Computational Mathematics 3, Elsevier Science Publishers B.V., 1992. MR 1172520 (93g:30007)

Lisa Lorentzen has changed her name from Lisa Jacobsen.

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