A Malmquist--Steinmetz theorem for difference equations

It is shown that if equation begin{equation*} f(z+1)^n=R(z,f), end{equation*} where $R(z,f)$ is rational in both arguments and $deg_f(R(z,f)) ot=n$, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term $R(z,f)$ takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case $deg_f(R(z,f))=n$ of the equation above and thus provide a complete difference analogue of Steinmetz generalization of Malmquists theorem. Finally, a description of how to simplify the classification in the case $deg_f(R(z,f))=n$ is given by using the new methods introduced in this paper.