On the relation between the continuous and discrete Painleve equations

A method for deriving difference equations (the discrete Painleve' equations in particular) from the Backlund transformations of the continuous Painleve' equations is discussed. This technique can be used to derive several of the known discrete Painleve' equations (in particular, the first and second discrete Painleve equations and some of their alternative versions). The Painleve' equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painleve' equations. Exact solutions of the Painleve equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.