Higher linearizable mappings and their explicit integration

We present results on linearizable mappings which can be obtained from discrete Painlevé equations associated with the affine Weyl groups E(1)8 and E(1)7 (hence the monicker 'higher') of the Sakai classification through limiting procedures. We derive their non-autonomous forms which involve an arbitrary function of the independent variable and present their explicit integration. We also obtain a relation between a sub-class of the mappings studied here and mappings of the Gambier type. We also show that the Gambier mappings, which are obtained as limits of discrete Painlevé equations, can be reduced to a single, 'master', form.