On a transcendental equation related to Painleve III, and its discrete forms

We examine the canonical forms of Painleve equations and argue that the equ ation for Pm in which one parameter is taken to be equal to zero should be considered as a canonical form different from the standard P-III Our argume nt is based on the fact that the value of this parameter cannot be modified through auto-Backlund transformations. We investigate the possible discret e forms of this equation and produce two of them. One is of a difference ty pe, where the independent variable enters linearly, while the second one is of q type where the independent variable enters in a multiplicative way. T he properties of these discrete equations are also studied.