From trihomographic to elliptic Painlevé equations

We use the trihomographic representation of discrete Painlevé equations in order to derive a new, factorised form for equations associated with the affine Weyl group $E_8^{\left(1\right)}$. Introducing an ancillary dependent variable we are able to obtain factorised forms for the general additive, multiplicative and elliptic discrete $E_8^{\left(1\right)}$-related Painlevé equations. The advantage of these factorised forms is that they make the singularity analysis of the $E_8^{\left(1\right)}$ straightforward in par with the situation for equations associated with lower affine Weyl groups. (letter)