Meromorphic solutions of some nonlinear difference equations of higher order (Q1074737)

The present paper deals with the question, whether or not the difference equation \[ (1)\quad \alpha_ ny(x+n)+\alpha_{n-1}y(x+n- 1)+...+\alpha_ 1y(x+1)=R(y(x)), \] R(y) rational over \({\mathbb{C}}\), \(\alpha_ 1,...,\alpha_ n\in {\mathbb{C}}\) admits a meromorphic solution which is not a constant). If \(R(y)=\frac{P(y)}{Q(y)}\) with the mutually prime polynomials P,Q of degree p,q respectively, it has been shown by the author [Arch. Ration. Mech. Anal. 91, 171-192 (1985; reviewed above)] that (1) admits nontrivial meromorphic solutions if \(p\geq q+2\). In the present paper the author shows, that this result remains even true if \(p\leq q+1\). Furthermore, the author gives expansions or asymptotic expansions of these solutions which for example in a special case are of the form \[ y(x)=c_{-1}\tau^ x+\sum^{\infty}_{k=m}c_ k\tau^{-kx} \] in \(D_{\rho}:=\{x|\) \(| \tau^{-x}| <\rho \}\).