Lens Generalisation of $τ$-functions for the Elliptic Discrete Painlevé Equation

We propose a new bilinear Hirota equation for $τ$-functions associated with the $E_8$ root lattice, that provides a "lens" generalisation of the $τ$-functions for the elliptic discrete Painlevé equation. Our equations are characterized by a positive integer $r$ in addition to the usual elliptic parameters, and involve a mixture of continuous variables with additional discrete variables, the latter taking values on the $E_8$ root lattice. We construct explicit $W(E_7)$-invariant hypergeometric solutions of this bilinear Hirota equation, which are given in terms of elliptic hypergeometric sum/integrals.

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