Authors’ abstract: R. Halburd
and R. Korhonen
[Existence of finite order meromorphic solutions as a detector of integrability in difference equations. (to appear)] have shown that the existence of sufficiently many meromorphic solutions of finite order is enough to single out a discrete form of the second Painlevé equation from a more general class
of complex difference equations. A key lemma in their reasoning is to show that
has to be of infinite order, provided that
and that a certain growth condition for the counting function of distinct poles of
holds. In this paper, we prove a generalization of this lemma to higher order difference equations of more general type. We also consider related complex functional equations.