Abstract
We study solutions (x(n))(n is an element of N) of nonhomogeneous nonlinear second order difference equations of the type
l(n) = x(n) (sigma(n,1) x(n+1) + sigma(n,0) x(n) + sigma(n,-1) x(n-1)) + kappa(n) x(n), n is an element of N,
with given initial data {x(0) is an element of R & x(1) is an element of R+}
where
(l(n))(n is an element of N) is an element of R+ & (sigma(n,0))(n is an element of N) is an element of R+ & (kappa(n))(n is an element of N) is an element of R,
and the left and right sigma-coefficients satisfy either
(sigma(n,1))(n is an element of N) is an element of R+ & (sigma(n,-1))(n is an element of N) is an element of R+
or
Depending on one's standpoint, such equations originate either from orthogonal polynomials associated with certain Shohat-Freud-type exponential weight functions or from Painleve's discrete equation #1, that is, d-P-I. (C) 2014 Elsevier Inc. All rights reserved.