arXiv subjects

Hidehito Nagao

Publications and source records attributed to Hidehito Nagao.

8 recordsLinked to original sources

arXiv2023

Three $q$-Painlevé equations with affine Weyl group symmetry of type $E_7^{(1)}$

Three $q$-Painlevé type equations are derived through degenerations of the $q$-Painlevé equation with affine Weyl group symmetry of type $q$-$E_8^{(1)}$. The three $q$-Painlevé type equations are associated with different realizations of the same symmetry/surface type $q$-$E_7^{(1)}$/$q$-$A_1^{(1)}$. We give three representations of affine Weyl group actions of type $q$-$E_7^{(1)}$, and relations among them. The three $q$-$E_7^{(1)}$ equations are also derived from the three representations respectively.

nlin.SI↗

arXiv2021

The Padé interpolation method applied to additive difference Painlevé equations

We study Padé interpolation problems on an additive grid, related to additive difference ($d$-) Painlevé equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By choosing suitable Padé problems, we can derive time evolution equations, scalar Lax pairs of contiguous type and determinant formulae of special solutions given in terms of hypergeometric functions, for the corresponding $d$-Painlevé equations.

nlin.SI↗

arXiv2017

A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$

Recently a certain $q$-Painlevé type system has been obtained from a reduction of the $q$-Garnier system. In this paper it is shown that the $q$-Painlevé type system is associated with another realization of the affine Weyl group symmetry of type $E_7^{(1)}$ and is different from the well-known $q$-Painlevé system of type $E_7^{(1)}$ from the point of view of evolution directions. We also study a connection between the $q$-Painlevé type system and the $q$-Painlevé system of type $E_7^{(1)}$. Furthermore determinant formulas of particular solutions for the $q$-Painlevé type system are constructed in terms of the terminating $q$-hypergeometric function.

nlin.SI↗

arXiv2017

Variations of $q$-Garnier system

We study several variants of q-Garnier system corresponding to various directions of discrete time evolutions. We also investigate a relation between the $q$-Garnier system and Suzuki's higher order $q$-Painlev/'e system by using a duality of the $q$-KP system.

nlin.SI↗

arXiv2016

Study of $q$-Garnier system by Padé method

We give a simple form of the evolution equation and a scalar Lax pair for the $q$-Garnier system. Some reductions to the $q$-Painlevé equations and the autonomous case as a generalized QRT system are discussed. Using two kinds of Padé problems on differential grid and $q$-grid, we derive some special solutions of the $q$-Garnier system in terms of the $q$-Appell Lauricella function and the generalized $q$-hypergeometric function.

nlin.SI↗

arXiv2016

The Padé interpolation method applied to $q$-Painlevé equations II (differential grid version)

Recently we studied Padé interpolation problems of $q$-grid, related to $q$-Painlevé equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_5^{(1)}$, $A_4^{(1)}$ and $(A_2+A_1)^{(1)}$. By solving those problems, we could derive evolution equations, scalar Lax pairs and determinant formulae of special solutions for the corresponding $q$-Painlevé equations. It is natural that the $q$-Painlevé equations were derived by the interpolation method of $q$-grid, but it may be interesting in terms of differential grid that the Padé interpolation method of differential grid (i.e. Padé approximation method) has been applied to the $q$-Painlevé equation of type $D_5^{(1)}$ by Y. Ikawa. In this paper we continue the above study and apply the Padé approximation method to the $q$-Painlevé equations of type $E_6^{(1)}$, $D_5^{(1)}$, $A_4^{(1)}$ and $(A_2+A_1)^{(1)}$. Moreover determinant formulae of the special solutions for $q$-Painlevé equation of type $E_6^{(1)}$ are given in terms of the terminating $q$-Appell Lauricella function.

math.CA↗

arXiv2016

Lax pairs for additive difference Painlevé equations

A Lax pair for the additive difference Painlevé equation of type $E_7^{(1)}$ is explicitly obtained as certain linear difference equations of scalar form. The compatibility of the Lax pair is proved by using certain characterization of the coefficients in the Lax equation. Some Lax pairs for types $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$ are also given by the degeneration.

nlin.SI↗

arXiv2015

The Padé interpolation method applied to $q$-Painlevé equations

We establish interpolation problems related to all the $q$-Painlevé equations of types from $E_7^{(1)}$ to $(A_2+A_1)^{(1)}$. By solving those problems, we can derive the evolution equations, the scalar Lax pairs and the determinant formulae of special solutions for the corresponding $q$-Painlevé equations.

math-ph↗