Existence of meromorphic solutions of first order difference equations

TL;DR

This paper classifies meromorphic solutions of certain first-order difference equations, showing they are either solutions to linear or Riccati equations or transform into a specific list of known equations with explicit elliptic function solutions.

Contribution

It provides a classification of meromorphic solutions for a class of first-order difference equations, extending Steinmetz's generalization of Malmquist's theorem.

Findings

Solutions are expressed in terms of elliptic functions or solutions to Riccati equations.

Meromorphic solutions are either linear, Riccati, or transform into a finite list of known equations.

The classification includes equations related to Fermat and QRT maps.

Abstract

It is shown that if It is shown that if \begin{equation}\label{abstract_eq} f(z+1)^n=R(z,f),\tag{\dag} \end{equation} where $R(z,f)$ is rational in $f$ with meromorphic coefficients and $\deg_f(R(z,f))=n$, has an admissible meromorphic solution, then either $f$ satisfies a difference linear or Riccati equation with meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if \eqref{abstract_eq}, where the assumption $\deg_f(R(z,f))=n$ has been discarded, has rational coefficients and a transcendental meromorphic solution $f$ of hyper-order $<1$, then either $f$ satisfies a difference linear or Riccati equation with rational coefficients, or \eqref{abstract_eq} can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions which are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz' generalization of Malmquist's theorem.