A new Clunie type theorem for difference polynomials

Abstract

It is shown that if w(z) is a finite-order meromorphic solution of the equation

where , , is a homogeneous difference polynomial with meromorphic coefficients, and and are polynomials in w(z) with meromorphic coefficients having no common factors such that

where ord₀(P) denotes the order of zero of at x ₀ = 0 with respect to the variable x ₀, then the Nevanlinna counting function N(r,w) satisfies . This implies that w(z) has a relatively large number of poles. For a smaller class of equations, a stronger assertion 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.

Keywords:

• Clunie theorem
• difference Riccati
• difference Painlevé equations
• value distribution

AMS Subject Classification::

• Primary: 39A10
• Secondary: 39A12, 30D35

Acknowledgements

The research reported in this paper was supported in part by the Academy of Finland grant #118314 and #210245. We would like to thank the anonymous referees for their helpful comments on the paper.