No abstract

We consider the q-Painlevé III equation arising from the birational representation of the affine Weyl group of type (A 2 + A 1 ) (1) . We study the reduction of the q-Painlevé III equation to the q-Painlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the τ functions.

We consider the symmetric q-Painlevé equations derived from the birational representation of affine Weyl groups by applying the projective reduction and construct the hypergeometric solutions. Moreover, we discuss continuous limits of the symmetric q-Painlevé equations to Painlevé equations together with their hypergeometric solutions.

We introduce the concept of ω-lattice, constructed from τ functions of Painlevé systems, on which quad-equations of ABS type appear. In particular, we consider the A (1) 5and A (1) 6 -surface q-Painlevé systems corresponding affine Weyl group symmetries are of (A 2 + A 1 ) (1) -and (A 1 + A 1 ) (1) -types, respectively.2010 Mathematics Subject Classification. 33E15, 33E17, 39A13, 39A14.

Abstract. We present a class of reductions of Möbius type for the lattice equations known as Q1, Q2, and Q3 from the ABS list. The deautonomised form of one particular reduction of Q3 is shown to exist on the A (1) 1 surface which belongs to the multiplicative type of rational surfaces in Sakai's classification of Painlevé systems. Using the growth of degrees of iterates, all other mappings that result from the class of reductions considered here are shown to be linearisable. Any possible linearisations are calculated explicitly by constructing a birational transformation defined by invariant curves in the blown up space of initial values for each reduction.

In this paper, we show how to relate n-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the 4-dimensional cube to the q-discrete third Painlevé equation, which is a dynamical system on a rational surface of type A(1) 5 with the extended affine Weyl group W (A2 + A1) (1) . We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A.

In this paper, we construct two lattices from the τ functions of -surface q-Painlevé equations, on which quad-equations of ABS type appear. Moreover, using the reduced hypercube structure, we obtain the Lax pairs of the -surface q-Painlevé equations.

The well known elliptic discrete Painlevé equation of Sakai is constructed by a standard translation on the lattice, given by nearest neighbor vectors. In this paper, we give a new elliptic discrete Painlevé equation obtained by translations along next-nearest-neighbor vectors. This equation is a generic (8-parameter) version of a 2-parameter elliptic difference equation found by reduction from Adler’s partial difference equation, the so-called Q4 equation. We also provide a projective reduction of the well known equation of Sakai.