Some Results on the Painlevé III Difference Equations with Constant Coefficients

In this paper, we investigate the following two Painleve III equations: $$\overline{w}\underline{w}(w^{2}-1)=w^{2}+\mu$$ and $$\overline{w}\underline{w}(w^{2}-1)=w^{2}-\lambda w$$ , where $$\overline{w}:=w(z+1)$$ , $$\underline{w}:=w(z-1)$$ and $$\mu$$ ( $$\mu\neq-1$$ ) and $$\lambda\notin\{\pm 1\}$$ are constants. We discuss the questions of existence of rational solutions, of Borel exceptional values and the exponents of convergence of zeros, poles and fixed points of transcendental meromorphic solutions of these equations.