Meromorphic solutions of difference Painlevé equations

We show that, under certain conditions, some difference Painlevé equations have nontrivial meromorphic solutions in the whole complex plane. These meromorphic solutions are obtained by analytic continuation of asymptotic solutions given in sectors of zero opening angle. The existence of these asymptotic solutions is shown by using fundamental matrix solutions of the associated linear systems with double unit characteristic roots and with coefficients expressed by asymptotic series in fractional powers.