Continued Fractions with Non-Integer Numerators
John Greene and Jesse Schmieg
Department of Mathematics and Statistics
University of Minnesota Duluth
Duluth, MN 55812
Anselm and Weintraub investigated a generalization of classic continued
fractions, where the
"numerator" 1 is replaced by an arbitrary positive
integer. Here, we generalize further to the case of an arbitrary real
number z ≥ 1. We focus mostly on the case where z
is rational but not an
integer. Extensive attention is given to periodic expansions and
expansions for √n,
where we note similarities and differences between
the case where z is an integer and when z is rational.
When z is not an
integer, it need no longer be the case that √n
has a periodic expansion.
We give several infinite families where periodic expansions of various
Full version: pdf,
Received September 18 2016; revised version received December 21 2016.
Published in Journal of Integer Sequences, December 23 2016.
Journal of Integer Sequences home page