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

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Abstract
We examine the canonical forms of Painlevé equations and argue that the equation for PIII in which one parameter is taken to be equal to zero should be considered as a canonical form different from the standard PIII . Our argument is based on the fact that the value of this parameter cannot be modified through auto-Bäcklund transformations. We investigate the possible discrete forms of this equation and produce two of them. One is of a difference type, where the independent variable enters linearly, while the second one is of q type where the independent variable enters in a multiplicative way. The properties of these discrete equations are also studied.

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