Existence of entire solution of some certain type difference equation (Q2839888)

In the paper under review, zero distribution of finite-order entire solutions of certain types of non-linear difference equations is studied. In addition, it is shown that the difference equation NEWLINE\[NEWLINE f(z)^n + f(z+1)^m = p(z) NEWLINE\]NEWLINE admits entire solutions of finite order only if \(m=n=1\) or \(m=n=2\). It is shown also that in the former case \(f^{(k+1)}(z)\) is periodic with period \(2\), and in the latter case \(p(z)\equiv p\) is a constant and NEWLINE\[NEWLINE f(z)=ae^{Az}+be^{-Az}, NEWLINE\]NEWLINE where \(a\), \(b\) and \(A\) are complex constants such that \(4ab=p\) and \(e^A=-i\).