Elliptic asymptotics in $q$-discrete Painlevé equations

Abstract

We study the asymptotic behaviour of two multiplicative-(q-) discrete Painlevé equations as their respective independent variable goes to infinity. It is shown that the generic asymptotic behaviours are given by elliptic functions. We extend the method of averaging to these equations to show that the energies are slowly varying. The Picard-Fuchs equation is derived for a special case of q-P III .