Some Results on the Painleve Ⅲ Difference Equations with Constant Coefficients

摘要

In this paper, we investigate the following two Painleve III equations: (w) over bar(w) under bar (w(2) - 1) = w(2) + mu and (w) over bar(w) under bar (w(2) - 1) = w(2) - lambda w, where (w) over bar := w(z + 1), (w) under bar := w(z - 1) and mu (mu not equal = -1) and lambda is not an element of {+/- 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.