## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, H. Konno, H. Sakai, J. Shiraishi, T. Suzuki and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2018), 1 - 65

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

#### 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**

Received: 4 April 2016

Revised: 10 August 2016

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1537499422

**Digital Object Identifier**

doi:10.2969/aspm/07610001

**Zentralblatt MATH identifier**

07039299

**Subjects**

Primary: 39A20: Multiplicative and other generalized difference equations, e.g. of Lyness type

Secondary: 33E17: Painlevé-type functions 33D70: Other basic hypergeometric functions and integrals in several variables

**Keywords**

elliptic Painlevé equation $E_8$ lattice $\tau$-function Hirota equation Casorati determinant elliptic hypergeometric integral

#### Citation

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