A new Clunie type theorem for difference polynomials

TL;DR

This paper establishes a new Clunie type theorem for difference polynomials, providing conditions under which solutions have many poles, and offers criteria to exclude certain finite-order Riccati solutions in difference equations.

Contribution

It introduces a novel Clunie type theorem for difference polynomials and applies it to characterize pole distributions and exclude specific Riccati solutions.

Findings

Solutions have many poles when conditions are met.

Stronger pole-density results for certain equations.

Necessary and sufficient conditions for solution patterns.

Abstract

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)) 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 Painleve type.