On meromorphic solutions of Malmquist type difference equations

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