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A new Clunie type theorem for difference polynomials. (English) Zbl 1213.39005
Summary: It is shown that if $w(z)$ is a finite-order meromorphic solution of the equation $$H(z,w)P(z,w)= Q(z,w),$$ where $P(z,w)= P(z,w(z),w(z+c_1),\dots,w(z+c_n))$, $c_1,\dots,c_n\in\Bbb C$, is a homogeneous difference polynomial with meromorphic coefficients, and $H(z,w)=H(z,w(z))$ and $Q(z,w)= Q(z,w(z))$ are polynomials in $w(z)$ with meromorphic coefficients having no common factors such that $$\max\{\deg_w(H),\deg_w(Q)-\deg_w(P)\}> \min\{\deg_w(P), \text{ord}_0(Q)\}- \text{ord}_0(P),$$ where $\text{ord}_0(P)$ denotes the order of zero of $P(z,x_0,x_1,\dots,x_n)$ 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 $w(z)$ 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 $w(z)$ is essentially as high as the growth of $w(z)$ 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.

39A10Additive difference equations
39A12Discrete version of topics in analysis
30D35Distribution of values (one complex variable); Nevanlinna theory
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