Contiguity relations for discrete and ultradiscrete Painlevé equations

We show that the solutions of ultradiscrete Painlevé equations satisfy contiguity relations just as their continuous and discrete counterparts. Our starting point are the relations for q-discrete Painlevé equations which we then proceed to ultradiscretise. In this paper we obtain results for the one-parameter q-PIII, the symmetric q-PIV and the q-PVI. These results show that there exists a perfect parallel between the properties of continuous, discrete and ultradiscrete Painlevé equations.