Bilinear structure and Schlesinger transforms of the $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$ equations

Details Bilinear structure and Schlesinger transforms of the $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$ equations Bilinear structure and Schlesinger transforms of the $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$ equations

1996-01-27

arxiv.org/abs/solv-int/9601004v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

10 pages, Plain TeX

Scientific paper

10.1016/0375-9601(96)00336-2

We show that the recently derived ($q$-) discrete form of the Painlev\'e VI equation can be related to the discrete P$_{\rm III}$, in particular if one uses the full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$.

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