Elliptic and multiplicative discrete Painlevé equations from the affine Weyl group E₈

Pursuing our study of discrete Painlevé equations associated with the affine Weyl group $E_8^{\left(1\right)}$, we present our results on multiplicative and elliptic equations. Our method is based on the trihomographic representation of discrete Painlevé equations, which was shown to be equally rich as the more customary representations. The relation of $E_8^{\left(1\right)}$ equations to ones associated with affine Weyl groups situated in the degeneration cascade of $E_8^{\left(1\right)}$ is also presented, since it constitutes a precious aid for the derivation of the $E_8^{\left(1\right)}$ systems. (fast track communication)