Abstract In this article we formulate a group of birational transformations that is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo–Sakai’s $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again: the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy, and a similarity reduction of the $q$-Drinfeld–Sokolov hierarchy.

In this article we propose an extension of Appell hypergeometric function F2 (or equivalently F3). It is derived from a particular solution of a higher order Painlevé system in two variables. On the other hand, an extension of Appell’s F1 was introduced by Tsuda. We also show that those two extensions are equivalent at the level of systems of linear partial differential equations.

In this article, we propose a class of six-dimensional Painlevé systems given as the monodromy preserving deformations of the Fuchsian systems. They are expressed as polynomial Hamiltonian systems of sixth order. We also discuss their particular solutions in terms of the hypergeometric functions defined by fourth order rigid systems.

The higher order Painlevé system of type D(1) 2n+2 wasproposed by Y. Sasano as an extension of PVI for the affine Weyl group symmetry with the aid of algebraic geometry for Okamoto initial value space. In this article, we give it as the monodromy preserving deformation of a Fuchsian system. © 2013 The authors.

We study the Drinfeld-Sokolov hierarchies of type An(1), associated with the regular conjugacy classes of W(An). A class of fourth order Painleve systems is derived from them by similarity reductions.

In a recent work, we proposed the coupled Painlevé VI system with A2n+1-(1) symmetry, which is a higher order generalization of the sixth Painlevé equation (PVI). In this article, we present its particular solution expressed in terms of the hypergeometric function n+1Fn. We also discuss a degeneration structure of the Painlevé system derived from the confluence of n+1Fn.

We present a new system of ordinary differential equations with affine Weyl group symmetry of type E(1)6. This system is expressed as a Hamiltonian system of sixth order with a coupled Painlevé VI Hamiltonian. © 2009 IOP Publishing Ltd.

A higher order Painleve system of type D-2n+2((1)) was introduced by Y. Sasano. It is an extension of the sixth Painleve equation (P-VI) for the affine Weyl group symmetry. It is also expressed as a Hamiltonian system of order 2n with a coupled Hamiltonian of PVI. In this article, we discuss a derivation of this system from a Drinfeld-Sokolov hierarchy.

The sixth Painlevé equation arises from a Drinfeld-Sokolov hierarchy of type D(1)4 by similarity reduction. © 2006 IOP Publishing Ltd.

We study the degenerate Garnier system which generalizes the fifth Painlevé equation PV. We present two classes of particular solutions, classical transcendental and algebraic ones. Their coalescence structure is also investigated. © 2006 IOP Publishing Ltd.

In this paper, we show that the Garnier system in n-variables has affine Weyl group symmetry of type B(1)n+3. We also formulate the τ-functions for the Garnier system (or the Schlesinger system of rank 2) on the root lattice Q(Cn+3) and show that they satisfy Toda equations, Hirota-Miwa equations and bilinear differential equations. © 2005, Division of Functional Equations, The Mathematical Society of Japan. All rights reserved.

本年度は主に次の二つの成果を得た． (a) 昨年度の研究では，拡大された可約アフィン・ワイル群の双有理表現のラックス形式が得られ，これによりq-ガルニエ系の一般化が系統的に定式化された．本年度はこの結果についての論文をまとめ，Math. Phys. Anal. Geom.誌に投稿し掲載された．また2021年9月の日本数学会や2022年3月のジョージア大学(アメリカ合衆国)での国際会議において，オンライン講演による研究発表を行った． (b) 研究代表者や長尾・山田による先行研究において，q-ガルニエ系はハイネのq-超幾何級数の比によって記述される特殊解を持つことが明らかにされていた．これと上記(a)の結果を組み合わせることで，拡大アフィン・ワイル群のq-超幾何級数への作用を具体的に記述することに成功した．具体的には，まずq-ガルニエ系の由来となる拡大アフィン・ワイル群において，極大な部分群で超幾何関数解の条件と両立するものを求め，その双有理表現の解関数への作用からq-超幾何級数への線形作用を導出し，得られた結果を正方行列を用いて系統的に記述した．これにより，q-超幾何級数の満たす線形差分方程式や昇降演算子の作用などが，すべてワイル群の平行移動変換として理解できるようになった． (b)の成果については，指導院生との共著論文として発表予定であり，現在論文準備中である．

まず、本年度出版された論文に関して概要を述べる。「研究発表」項目第1番目の論文では、大きなパラメータを持つ超幾何微分方程式に対してヴォロス係数を定義し、一般的な形で明示式を与えた。2番目の論文では、大きなパラメータを持つ超幾何関数・合流型超幾何関数と、これらが満たす微分方程式のWKB解のボレル和との関係を一般的な場合に完全に記述した。応用として、これらの関数のパラメータに関する漸近展開公式を与えた。3番目の論文では、一般化されたq-ガルニエ系に対するLax形式を与えた。本年度実施した研究では、次のような成果が得られた。一般化超幾何関微分方程式に対する原点および無限遠点におけるヴォロス係数を定義し，その具体形を与えた。さらに、一般化超幾何微分方程式の間に成り立つ退化図式と、ヴォロス係数に対する極限操作が整合していることを見出した。多変数の場合への研究にも着手した。その原型として、大きなパラメータを持つエアリーの微分方程式のWKB解をホロノミック系の立場から見直し、接続公式の初等的な別証明を与えた。従来の標準的理論では、超幾何関数の接続公式を用いていた証明に対して、新たに得られた証明は、代数関数の接続のみを用いる。この立場から、多変数の場合の最も基本的なものとして大きなパラメータを持つパーシー積分が満たすホロノミック系（パーシー系と呼ぶ）について研究を行い、エアリーの微分方程式と同様の構造があることを見出した。これにより、パーシー系のWKB解のリサージェンスが証明できた。これらの成果を記載した論文は、現在執筆中である。

Recently higher order generalizations of the Painleve equations are proposed both on continuous side and on discrete side. However we haven't clarified yet a classification of equations or a relationship with hypergeometric functions. In this work we have formulated birational representations of a reducible extended affine Weyl group with the aid of cluster mutations. Translations of this group provide the known higher order q-Painleve equations containing the q-hypergeometric functions as particular solutions.

Introducing a large parameter in the 3 parameters contained in the Gauss hypergeometric differential equation, we can construct the WKB solutions which are formal solutions to the equation. The construction is done algebraically and elementarily, however, these formal solutions are divergent in general and do not have analytic sense. We may apply the Borel resummation method to the formal solutions and can construct analytic solutions and bases of the solution space. On the other hand, the Gauss hypergeometric differential equation has standard bases of solutions expressed by the hypergeometric function. In this research, we have obtained linear relations between these two classes of bases. As an application, asymptotic expansion formulas with respect to the large parameter of the Gauss hypergeometric function have been obtained. At the same time, we have some formulas which describe the parametric Stokes phenomena of the WKB solutions.

We study a q-analogue of the Painleve equation, which is a difference equation. The main subject is a study of local behavior of generic solutions of q-Painleve equations around a fixed singular point. In case of nonlinear q-difference equation, it is difficult to express local behavior of solutions. We solve a connection problem of some linear q-difference equations with irregular singular points at first. Then we express local behavior of generic solutions of q-Painleve equations around the origin in terms of connection coefficients of the linear q-difference equation. In cases of degenerated q-Painleve equations, the corresponding linear equation has an irregular singular point. Since some local solutions are represented by divergent series, the Stokes phenomenon appears. We decided the Stokes coefficients by means of connection coefficients of q-hypergeometric equations.

In this research, we have investigated the global properties of solutions to differential equations with a large parameter from the view point of the exact WKB analysis. There are three main results. Firstly, we have constructed the exponential-asymptotic (instanton-type) solutions, namely general formal solutions, to the equations which belong to the first Painleve hierarchies. Secondly, we have classified the topological types of the Stokes curves of the Gauss equation in terms of the parameters of the equation. Thirdly we have defined and computed explicit forms of the Voros coefficients of Gauss equation with a large parameter and obtained the Borel sums go them. We have obtained the formulas that describe parametric Stokes phenomena of WKB solutions.