For integers m≥2, we study divergent continued fractions whose
numerators and denominators in each of the m arithmetic progressions modulo
m converge. Special cases give, among other things, an infinite sequence of
divergence theorems, the first of which is the classical Stern-Stolz theorem.
We give a theorem on a general class of Poincar{\'e} type recurrences which
shows that they tend to limits when the limits are taken in residue classes and
the roots of their characteristic polynomials are distinct roots of unity.
We also generalize a curious q-continued fraction of Ramanujan's with three
limits to a continued fraction with k distinct limit points, k≥2. The
k limits are evaluated in terms of ratios of certain unusual q series.
Finally, we show how to use Daniel Bernoulli's continued fraction in an
elementary way to create analytic continued fractions with m limit points,
for any positive integer m≥2.Comment: 29 pages. Updated/new conten