Lens Generalisation of $\tau$-functions for the Elliptic Discrete Painlev\'e Equation

TL;DR

This paper introduces a new bilinear Hirota equation for $ au$-functions on the $E_8$ lattice, generalizing elliptic discrete Painlevé equations with explicit solutions involving elliptic hypergeometric functions.

Contribution

It presents a novel lens generalization of $ au$-functions incorporating an integer parameter and mixed variables, expanding the mathematical framework of elliptic Painlevé equations.

Findings

Derived a new bilinear Hirota equation for $E_8$ $ au$-functions.

Constructed explicit $W(E_7)$-invariant elliptic hypergeometric solutions.

Established a connection between discrete lattice variables and continuous elliptic parameters.

Abstract

We propose a new bilinear Hirota equation for $\tau$-functions associated with the $E_8$ root lattice, that provides a "lens" generalisation of the $\tau$-functions for the elliptic discrete Painlev\'e 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.