Divergence of continued fractions related to hypergeometric series

Abstract

Let K ( a n / b n ) K({a_n}/{b_n}) be a limit periodic continued fraction of elliptic type; i.e., a n → a {a_n} \to a and b n → b {b_n} \to b , where a / ( b + w ) a/(b + w) is an elliptic linear fractional transformation of w. We show that if ∑ | a n − a | > ∞ \sum {|{a_n} - a| > \infty } and ∑ | b n − b | > ∞ \sum {|{b_n} - b| > \infty } , then K ( a n / b n ) 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 = O ( n − 1 ) {a_n} - a = \mathcal {O}({n^{ - 1}}) and b n = b = 1 {b_n} = b = 1 and K ( a n / b n ) K({a_n}/{b_n}) converges. The divergence result is also generalized further.