Discrete systems related to some equations of the Painlev\'e-Gambier classification

Abstract

We derive integrable discrete systems which are contiguity relations of two equations in the Painlev\'e-Gambier classification depending on some parameter. These studies extend earlier work where the contiguity relations for the six transcendental Painlev\'e equations were obtained. In the case of the Gambier equation we give the contiguity relations for both the continuous and the discrete system.