On the Convergence of Limit Periodic Continued Fractions K(an/1) where an→ - ¼. Part IV |
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Authors: | Lisa Lorentzen |
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Institution: | (1) Department of Mathematics Norwegian University of Science and Technology N—7491 Trondheim Norway, NO |
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Abstract: | Continued fractions K(a
n
/b
n
) , where a
n
, b
n
∈\smallbf C and a
n
/b
n
b
n-1
→-\frac 14 , may converge or diverge depending on how a
n
/b
n
b
n-1
approaches its limit. Due to equivalence transformations it suffices to study the special case where all b
n
=1 . We shall prove that K(a
n
/1) converges if a
n
→-\frac 14 and there exists a set V\subseteq\smallbf C \cup{∈fty} with certain properties such that a
n
/(1+V)\subseteq V for all n . We shall also summarize some other useful consequences of such value sets V .
January 31, 2000. Date revised: July 28, 2000. Date accepted: August 16, 2000. |
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Keywords: | , Parabolic limit periodic continued fractions, Convergence, Divergence, Value sets, AMS Classification, 40A15, |
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