On the relation between the continuous and discrete Painleve equations

A method for deriving difference equations (the discrete Painleve' equation s in particular) from the Backlund transformations of the continuous Painle ve' equations is discussed. This technique can be used to derive several of the known discrete Painleve' equations (in particular, the first and secon d discrete Painleve equations and some of their alternative versions). The Painleve' equations possess hierarchies of rational solutions and one-param eter families of solutions expressible in terms of the classical special fu nctions for special values of the parameters. Hence, the aforementioned rel ations can be used to generate hierarchies of exact solutions for the assoc iated discrete Painleve' equations. Exact solutions of the Painleve equatio ns simultaneously satisfy both a differential equation and a difference equ ation, analogously to the special functions.