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

N1 - Funding Information:
Data accessibility. This paper does not contain any additional data. Authors’ contributions. All authors contributed equally to writing the paper. Competing interests. We have no competing interests. Funding. This research was supported by an Australian Laureate Fellowship no. FL120100094 and grant no. DP160101728 from the Australian Research Council and JSPS KAKENHI grant nos. JP16H03941, JP16K13763 and JP17J00092.

PY - 2017/11/1

Y1 - 2017/11/1

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.

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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 A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2207

M1 - 20170312

ER -