Abstract
It is shown that if w(z) is a finite-order meromorphic solution of the equation
where
![](//:0)
,
![](//:0)
, is a homogeneous difference polynomial with meromorphic coefficients, and
![](//:0)
and
![](//:0)
are polynomials in
w(
z) with meromorphic coefficients having no common factors such that
where ord
0(
P) denotes the order of zero of
![](//:0)
at
x
0 = 0 with respect to the variable
x
0, then the Nevanlinna counting function
N(
r,
w) satisfies
![](//:0)
. This implies that
w(
z) has a relatively large number of poles. For a smaller class of equations, a stronger assertion
![](//:0)
is obtained, which means that the pole density of
w(
z) is essentially as high as the growth of
w(
z) allows. As an application, a simple necessary and sufficient condition is given in terms of the value distribution pattern of the solution, which can be used as a tool in ruling out the possible existence of special finite-order Riccati solutions within a large class of difference equations containing several known difference equations considered to be of Painlevé type.
AMS Subject Classification::
Acknowledgements
The research reported in this paper was supported in part by the Academy of Finland grant #118314 and #210245. We would like to thank the anonymous referees for their helpful comments on the paper.