Some Properties of Solutions for Someq-Difference Equations Containing Painlevé Equation

Abstract

The existence and growth of meromorphic solutions f(z) for some q-difference equations are studied, and some estimates for the exponent of convergence of poles of Δqf, Δq2f, Δqf/f, and Δq2f/f are also obtained. Our theorems are improvements and extensions of the previous results.

On existence of meromorphic solutions for nonlinear q-difference equation

“…Thus, for each sufficiently large s, there exists a j ∈ such that s ∈ [|q| j r 0 , |q| j+1 r 0 ), i.e., j > log s−log(|q|r 0 ) log |q| . Therefore, (15)…”

“…Recently, some authors investigated zero order meromorphic solutions of q-difference equations [8,11,14,15]. Qi and Yang [13] considered q-difference Painlevé equation I, and obtained the following Theorem 8.…”

“…These equations are now known as the q-difference analogues of difference Painlevé equations I and II. Some results about transcendental meromorphic solutions of zero order to (4)-( 6), can be found in [13][14][15].…”