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A q-analogue of the Drinfeld-Sokolov hierarchy of type A and a higher order generalization of the $q$-Painlevé VI equation

Hold Date
2011-12-05 15:30〜2011-12-05 17:00
Seminar Room 7, Faculty of Mathematics building, Ito Campus
Object person
Takao SUZUKI (Osaka Prefecture University)

The Drinfeld-Sokolov hierarchies are extensions of the KP (or mKP)
hierarchy for the affine Lie algebras.  And they imply several
Painlev¥'{e} type differential equations by similarity reductions.
For type $A$ among them, the coupled Painlev¥'{e} VI system is
derived, which admits a particular solution in terms of the
generalized hypergeometric function.  In this talk, we consider its
$q$-analogue.  Namely, we formulate a $q$-DS hierarchy of type $A$ and
derive a $q$-Painlev¥'{e} system of $2n$-th order by a similarity
reduction.  Such a $q$-Painlev¥'{e} system contains the
$q$-Painlev¥'{e} VI equation proposed by Jimbo and Sakai as the case
$n=1$.  We can show this fact with the aid of a $q$-Laplace
transformation for a system of linear $q$-difference equations.