Existence of meromorphic solutions of first-order difference equations (Q2187720)

The authors prove that if \(f(z + 1)^{n} = R(z, f)\), where \(R(z, f)\) is rational in \(f\) with meromorphic coefficients and \(\deg_{f} R(z, f) = n\), has an admissible meromorphic solution \(f\), then either \(f\) satisfies a difference linear or Riccati equation with meromorphic coefficients or \(f(z + 1)^{n} = R(z, f)\) can be transformed into one of ten specific equations. It is also proved that if the hypothesis \(\deg_{f} R(z, f) = n\) is removed and \(f(z + 1)^{n} = R(z, f)\) with rational coefficients has a transcendental meromorphic solution \(f\) of hyper-order less than \(1\), then \(f\) satisfies a difference linear or Riccati equation or \(f(z + 1)^{n} = R(z, f)\) can be transformed into one of five specific equations.