ON THE IDENTITY OF TWO q-DISCRETE PAINLEVÉ EQUATIONS AND THEIR GEOMETRICAL DERIVATION

One of the characteristics of discrete Painlevé equations is that they may possess more than one canonical form. Indeed we often encounter equations which are written as a system involving several dependent variables. Since by definition the discrete Painlevé equations are second-order mappings, these multicomponent systems include equations which are local. It is then straightforward, if some equation is linear in one of the variables, to solve for this variable and eliminate it from the final system. One thus obtains two perfectly equivalent forms which may have totally different aspects. This feature is in contrast with the continuous Painlevé case where the latitude left by the transformations which preserve the Painlevé property is minimal. The fact that there exist just 6 continuous Painlevé equations at second order while the number of possible second-order discrete Painlevé equations is in principle infinite may play a role. The possible existence of an unlimited number of discrete Painlevé equations has been explicitly pointed out in [2]. In that paper we have, in fact, presented a novel definition for the discrete Painlevé equations. The traditional definition of a discrete Painlevé equation is that of an integrable, nonautonomous, second-order mapping, the continuous limit of which is a continuous Painlevé equation. This definition turned out to be severely limitative since it binds the discrete systems to the continuous ones through the continuous limit. However, as was shown repeatedly, the discrete systems are more fundamental than their continuous counterparts, and in the case of discrete Painlevé equations much richer, as far as the degrees of freedom are concerned. We were thus naturally led in [2] to propose