On Finite-Order Meromorphic Solutions of a Class Containing a Difference Painlevé I Equation

Abstract

In this article, we investigate the growth and value distribution of finite-order transcendental meromorphic solutions for some non-linear difference equations. For instance, we show that demanding the existence of a transcendental meromorphic solution of a large class of difference equations containing difference Painleve I reduces the class either to a smaller one which still contains the difference Painleve I, or to a class of first-order equations. In addition, for certain classes of equations, we prove the existence of rational solutions and give their forms.