*(English)*Zbl 1213.39005

Summary: It is shown that if $w\left(z\right)$ is a finite-order meromorphic solution of the equation

where $P(z,w)=P(z,w\left(z\right),w(z+{c}_{1}),\cdots ,w(z+{c}_{n}))$, ${c}_{1},\cdots ,{c}_{n}\in \u2102$, is a homogeneous difference polynomial with meromorphic coefficients, and $H(z,w)=H(z,w(z\left)\right)$ and $Q(z,w)=Q(z,w(z\left)\right)$ are polynomials in $w\left(z\right)$ with meromorphic coefficients having no common factors such that

where ${\text{ord}}_{0}\left(P\right)$ denotes the order of zero of $P(z,{x}_{0},{x}_{1},\cdots ,{x}_{n})$ at ${x}_{0}=0$ with respect to the variable ${x}_{0}$, then the Nevanlinna counting function $N(r,w)$ satisfies $N(r,w)\ne S(r,w)$. This implies that $w\left(z\right)$ has a relatively large number of poles. For a smaller class of equations, a stronger assertion $N(r,w)=T(r,w)+S(r,w)$ is obtained, which means that the pole density of $w\left(z\right)$ is essentially as high as the growth of $w\left(z\right)$ allows. As an application, a simple necessary and sufficient condition is given in terms of the value distribution pattern of the solution, which can be used as a tool in ruling out the possible existence of special finite-order Riccati solutions within a large class of difference equations containing several known difference equations considered to be of Painlevé type.