## Advanced Studies in Pure Mathematics

### Remarks on $\tau$-functions for the difference Painlevé equations of type $E_8$

Masatoshi Noumi

#### Abstract

We investigate the structure of $\tau$-functions for the elliptic difference Painlevé equation of type $E_8$. Introducing the notion of ORG $\tau$-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 $\tau$-functions associated with the configuration of generic nine points in the projective plane.

#### Article information

Dates
Revised: 10 August 2016
First available in Project Euclid: 21 September 2018

Noumi, Masatoshi. Remarks on $\tau$-functions for the difference Painlevé equations of type $E_8$. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 1--65, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610001. https://projecteuclid.org/euclid.aspm/1537499422