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Mathematics of Computation

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Divergence of continued fractions related to hypergeometric series

Author: Lisa Lorentzen
Journal: Math. Comp. 62 (1994), 671-686
MSC: Primary 40A15; Secondary 33C05
MathSciNet review: 1203736
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Abstract: Let $ K({a_n}/{b_n})$ be a limit periodic continued fraction of elliptic type; i.e., $ {a_n} \to a$ and $ {b_n} \to b$, where $ a/(b + w)$ is an elliptic linear fractional transformation of w. We show that if $ \sum {\vert{a_n} - a\vert < \infty } $ and $ \sum {\vert{b_n} - b\vert < \infty } $, then $ K({a_n}/{b_n})$ diverges. This generalizes the well-known Stern-Stolz Theorem. The Gauss continued fraction (related to hypergeometric functions) is used as an example. We also give an example where $ {a_n} - a = \mathcal{O}({n^{ - 1}})$ and $ {b_n} = b = 1$ and $ K({a_n}/{b_n})$ converges. The divergence result is also generalized further.

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Keywords: Hypergeometric series, divergence of continued fractions
Article copyright: © Copyright 1994 American Mathematical Society

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