New Expressions for Discrete Painlev\'e Equations

Abstract

It is known that discrete Painlev\'e equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlev\'e equations in which the symmetries become clearly visible. We know how to obtain discrete Painlev\'e equations from certain rational surfaces in connection with the extended affine Weyl groups. By means of this representation, we clarify the relation between the equation and the surface.