Existence of Meromorphic Solutions of First-Order Difference Equations

Abstract

It is shown that if f(z+1)n=R(z,f), where R(z, f) is rational in f with meromorphic coefficients and degf(R(z,f))=n, has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if f(z+1)n=R(z,f), where the assumption degf(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 the equation above 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 that are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz’ generalization of Malmquist’s theorem.