Discrete Painleve equations: coalescences, limits and degeneracies

Abstract

Starting from the standard form of the five discrete Painlev\'e equations we show how one can obtain (through appropriate limits) a host of new equations which are also the discrete analogues of the continuous Painlev\'e equations. A particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new $n$-dependence and introduces more new discrete Painlev\'e equations.