Meromorphic solutions of Painlevé III difference equations with Borel exceptional values

In this paper, we investigate the properties of meromorphic solutions of Painlevé III difference equations. In particular, the difference equation w¯w̲w(w−1)=μ with μ being a non-zero constant is studied. We show that the rational solutions of the equation assume only one form and the transcendental solutions have at most one Borel exceptional value. We also show that the difference equation w¯w̲(w−1)2=(w−λ)2 does not have nonconstant rational solution, where λ (≠0,1) is a constant.