We present a new, nonautonomous Lax pair for a lattice nonautomous modified Korteweg-deVries equation and show that it can be consistently extended multidimensionally, a property commonly referred to as being consistent around a cube. This nonautonomous equation is reduced to a series of q-discrete Painlevé equations, and Lax pairs for the reduced equations are found. A 2 × 2 Lax pair is given for a qP III with multiple parameters and, also, for versions of qP II and qP V , all for the first time.

In this paper we discuss the integrability properties of a nonlinear partial difference equation on the square obtained by the multiple scale integrability test from a class of multilinear dispersive equations defined on a four points lattice.

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.

Abstract. We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric t functions for the sixth Painlevé equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface. Moreover, we show that the discrete power function is an immersion when the real part of the exponent is equal to one.

We present, for the first time, two hierarchies of nonlinear, integrable q-difference equations, one of which includes a q-difference form of each of the second and fifth Painlevé equations, qP II and qP V , the other includes qP III. All the equations have multiple free parameters. A method to calculate a 2 × 2 Lax pair for each equation in the hierarchy is also given.