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Continued fractions with multiple limits
- Source :
- Advances in Mathematics. (2):578-606
- Publisher :
- Elsevier Inc.
-
Abstract
- For integers $m \geq 2$, 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, $k\geq 2$. 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 $m \geq 2$.<br />Comment: 29 pages. Updated/new content
- Subjects :
- Mathematics(all)
Root of unity
General Mathematics
q-Series
010103 numerical & computational mathematics
01 natural sciences
Ramanujan's sum
Limit periodic continued fractions
symbols.namesake
Integer
q-Continued fractions
FOS: Mathematics
Periodic continued fraction
Number Theory (math.NT)
0101 mathematics
Continued fraction
Mathematics
Discrete mathematics
Mathematics - Number Theory
010102 general mathematics
Euler's continued fraction formula
16. Peace & justice
11A55
symbols
Chain sequence
Poincaré-type recurrences
Generalized continued fraction
Subjects
Details
- Language :
- English
- ISSN :
- 00018708
- Issue :
- 2
- Database :
- OpenAIRE
- Journal :
- Advances in Mathematics
- Accession number :
- edsair.doi.dedup.....c570c50ccee728b9a284de25415a89d5
- Full Text :
- https://doi.org/10.1016/j.aim.2006.07.004