Remarks on $τ$-functions for the difference Painlevé equations of type $E_8$

We investigate the structure of $τ$-functions for the elliptic difference Painlevé equation of type $E_8$. Introducing the notion of ORG $τ$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $τ$-functions associated with the configuration of generic nine points in the projective plane.

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