Polynomial difference equations which have entire solutions of finite order (Q1067120)

It is the aim of this note to determine those equations from the class \[ y(x+1)^ m=a_ p y(x)^ p+a_{p-1} y(x)^{p-1}+...+a_ 1 y(x)+a_ 0\quad, \] \(a_ p,a_{p-1},...,a_ 0\) are constants, \(a_ p\neq 0\), \(m\in {\mathbb{N}}\), which admit nontrivial entire solutions of finite order. The resulting equations are \(y(x+1)^ 2=A^ 2-y(x)^ 2\), \(A\neq 0\) and \(y(x+1)^ m=(ay(x)+b)^ m\), \(m\in {\mathbb{N}}\).