### Irreducibility of q-Painlevé equation of type $A_6^{(1)}$ in the sense of order

Nishioka, S *(Tokyo)*

Wednesday 13 May 2009, 17:00-17:30

Satellite

#### Abstract

I introduce a result on the irreducibility of q-Painlevé equation of type $A_6^{(1)}$ in the sense of order using the notion of decomposable extensions. The equation is one of the special non-linear q-difference equations of order 2 with symmetry $(A_1+A_1)^{(1)}$ and is also called q-Painlevé equation of type II.
The decomposable difference field extension is a difference analogue of K. Nishioka's which was defined to prove the irreducibility of the first Painlevé equation in the sense of Nishioka-Umemura. The strongly normal extension of difference fields defined by Bialynicki-Birula is decomposable. I proved that transcendental solutions of the equation in a decomposable extension may exist only for special parameters, and that all of them satisfies the identical well-known Riccati equation if we apply the Bäcklund transformations to it appropriate times.

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