On the extension of the Painleve property to difference equations

It is well known that the integrability (solvability) of a differential equ ation is related to the singularity structure of its solutions in the compl ex domain-an observation that lies behind the Painleve test. A number of wa ys of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural inter pretation of these equations in the complex domain as difference or delay e quations is described and it is noted that arbitrary periodic functions pla y an analogous role for difference equations to that played by arbitrary co nstants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the an alysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which s how that, module these periodic functions, solutions of a large class of di fference equations are meromorphic, regardless of their integrability. It i s argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integra bility in a large class of equations. A perturbative method is then constru cted and used to develop series in z and the derivative of log Gamma(z), wh ere z is the independent variable of the difference equation. This method p rovides an analogue of the series developed in the Painleve test for differ ential equations. Finally, the implications of these observations are discu ssed for two tests which have been studied in the literature regarding the integrability of discrete equations.