On meromorphic solutions of Malmquist type difference equations

Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form \(f(z+1)^n=R(z,f)\) \((\dagger)\) that have transcendental meromorphic solutions, where \(R(z,f)\) is rational in both arguments. In this paper, we first complete the classification for the case \(\deg_{f}(R(z,f))=n\) of \((\dagger)\) by identifying a new equation that was left out in our previous work. We will actually derive all the equations in this case based on some new observations on \((\dagger)\). Then, we study the relations between \((\dagger)\) and its differential counterpart \((f')^n=R(z,f)\). We show that most autonomous equations, singled out from \((\dagger)\) with \(n=2\), have a natural continuum limit to either the differential Riccati equation \(f'=a+f^2\) or the differential equation \((f')^2=a(f^2-\tau_1^2)(f^2-\tau_2^2)\), where \(a\neq 0\) and \(\tau_i\) are constants such that \(\tau_1^2\neq\tau_2^2\). The latter second degree differential equation and the symmetric QRT map are derived from each other using the bilinear method and the continuum limit method.