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

Nalini Joshi; Kenji Kajiwara; Tetsu Masuda; Nobutaka Nakazono; Yang Shi

doi:10.1098/rspa.2017.0312

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

Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono, Yang Shi

Division of Applied Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

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 P_VI, 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.

Original language	English
Article number	20170312
Journal	Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume	473
Issue number	2207
DOIs	https://doi.org/10.1098/rspa.2017.0312
Publication status	Published - Nov 1 2017

All Science Journal Classification (ASJC) codes

Mathematics(all)
Engineering(all)
Physics and Astronomy(all)

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title = "Geometric description of a discrete power function associated with the sixth Painlev{\'e} equation",

abstract = "In this paper, we consider the discrete power function associated with the sixth Painlev{\'e} 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.",

author = "Nalini Joshi and Kenji Kajiwara and Tetsu Masuda and Nobutaka Nakazono and Yang Shi",

note = "Funding Information: Data accessibility. This paper does not contain any additional data. Authors{\textquoteright} 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. Publisher Copyright: {\textcopyright} 2017 The Author(s) Published by the Royal Society. All rights reserved.",

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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. Publisher Copyright: © 2017 The Author(s) Published by the Royal Society. All rights reserved.

PY - 2017/11/1

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

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