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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 ),,w(z+c n )), c 1 ,,c n , 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),ord 0 (Q)}-ord 0 (P),

where ord 0 (P) denotes the order of zero of P(z,x 0 ,x 1 ,,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)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.

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