Divergence of continued fractions related to hypergeometric series

Lorentzen, Lisa

doi:10.1090/S0025-5718-1994-1203736-3

Divergence of continued fractions related to hypergeometric series
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Math. Comp. 62 (1994), 671-686 Request permission

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 $\sum {|{a_n} - a| < \infty }$ and $\sum {|{b_n} - b| < \infty }$, 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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