TY - JOUR

T1 - Geometric description of a discrete power function associated with the sixth Painlevé equation

AU - Joshi, Nalini

AU - Kajiwara, Kenji

AU - Masuda, Tetsu

AU - Nakazono, Nobutaka

AU - Shi, Yang

PY - 2017/11/22

Y1 - 2017/11/22

N2 - In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with W (3A(1) 1 ) symmetry. By constructing the action of W (3A(1) 1 ) as a subgroup of W (D(1) 4 ), i.e. the symmetry group of PVI, we show how to relate W (D(1) 4 ) to the symmetry group of the lattice. Moreover, by using translations in W (3A(1) 1 ), we explain the odd–even structure appearing in previously known explicit formulae in terms of the t function.

AB - In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with W (3A(1) 1 ) symmetry. By constructing the action of W (3A(1) 1 ) as a subgroup of W (D(1) 4 ), i.e. the symmetry group of PVI, we show how to relate W (D(1) 4 ) to the symmetry group of the lattice. Moreover, by using translations in W (3A(1) 1 ), we explain the odd–even structure appearing in previously known explicit formulae in terms of the t function.

KW - ? function

KW - ABS equation

KW - Affine Weyl group

KW - Discrete power function

KW - Painlevé equation

KW - Projective reduction

UR - http://www.scopus.com/inward/record.url?scp=85037720560&partnerID=8YFLogxK

UR - http://purl.org/au-research/grants/arc/DP160101728

UR - http://purl.org/au-research/grants/arc/FL120100094

U2 - 10.1098/rspa.2017.0312

DO - 10.1098/rspa.2017.0312

M3 - Article

AN - SCOPUS:85037720560

VL - 473

JO - PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES

JF - PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES

SN - 1364-5021

IS - 2207

M1 - 20170312

ER -