DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

PRODUCT DIFFERENCE EQUATIONS APPROXIMATING
RATIONAL EQUATIONS
KENNETH S. BERENHAUT AND JOHN D. FOLEY
We introduce a family of recursive sequences, involving products which, for certain initial
values, approximate some heavily studied rational equations. Some of the structure
of solutions which holds for the rational equation but not for the associated linearized
equation appear to be satisfied for the product approximation. Convergence of solutions
for one particular second-order member of the family is proved.
Copyright © 2006 K. S. Berenhaut and J. D. Foley. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly
cited.
1. Introduction
Solutions to rational difference equations of the form
y n = A + y n−k
y n−m
, (1.1)
for n ≥ 1, with k,m ∈{1,2,...}, have been studied extensively in recent years (cf. [1–7],
and the references therein).
Setting z n = y n − (A +1),(1.1)canberewrittenintheform
z n = z n−k − z n−m
A +1+z n−m
. (1.2)
We then have for |z n−m |