A Malmquist--Steinmetz theorem for difference equations

TL;DR

This paper extends Malmquist--Steinmetz theorem to certain difference equations, classifying solutions with transcendental meromorphic functions and linking them to elliptic and exponential functions.

Contribution

It provides a complete difference analogue of Steinmetz' generalization of Malmquist's theorem for equations with rational terms and specific degree conditions.

Findings

Solutions are expressed via elliptic and exponential functions.

Classifies difference equations with transcendental meromorphic solutions.

Completes the difference analogue of Steinmetz' generalization.

Abstract

It is shown that if the equation \begin{equation*} f(z+1)^n=R(z,f), \end{equation*} where $R(z,f)$ is rational in both arguments and $\deg_f(R(z,f))\not=n$, 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 Jacobian 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 Malmquist's theorem.