Document Type

Article

Publication Date

2007

Abstract

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 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 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.

Publication Title

Advances in Mathematics

ISSN

0001-8708

Publisher

Elsevier

Volume

210

Issue

2

First Page

578

Last Page

606

Comments

Preprint version is available here.

Download

Included in

Number Theory Commons

Share

COinS