A Malmquist–Steinmetz Theorem for Difference Equations

Abstract It is shown that if the equation $$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$ f ( z + 1 ) n = R , where R ( z , f ) rational in both arguments and $$\deg _f(R(z,f))\not =n$$ deg ≠ has a transcendental meromorphic solution, then above reduces into one out of several types difference equations term takes particular forms. Solutions these are presented terms Weierstrass or Jacobian elliptic functions, exponential type functions which solutions to certain autonomous first-order having with preassigned asymptotic behavior. These results complement our previous work on case _f(R(z,f))=n$$ thus provide complete analogue Steinmetz’ generalization Malmquist’s theorem.