DISCRETE VERSIONS OF THE PAINLEVE EQUATIONS

We present discrete forms of the Painleve transcendental equations P(III), P(IV), and P(V) that complement the list of the already known P(I) and P(II). These, most likely integrable, nonautonomous mappings go over to the usual Painleve equations in the continuous limit, while in the autonomous limit we recover discrete systems that belong to the integrable family of Quispel et al. Finally, we show that the discrete Painleve mappings satisfy the same reduction relations as the continuous Painleve transcendents, namely, P(V) --> {P(III),P(IV)} --> P(II) --> P(I).