
Paper

HermitePadé approximation, isomonodromic deformation and hypergeometric integral (with T. Mano)
Math. Z., in press.
We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by blockToeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painlevé equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

On a fundamental system of solutions of a certain hypergeometric equation
Ramanujan J. 38 (2015), 597618
We study the linear Pfaffian systems satisfied by a certain class of hypergeometric functions, which includes Gauß's {}_2F_1, Thomae's {}_LF_{L1} and AppellLauricella's F_D. In particular, we present a fundamental system of solutions with a characteristic local behavior by means of Eulertype integral representations. We also discuss how they are related to the theory of isomonodromic deformations or Painlevé equations.

Hypergeometric solution of a certain polynomial Hamiltonian system of isomonodromy type
Quart. J. Math. 63 (2012), 489505
In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painlevé equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauß's hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.

UC hierarchy and monodromy preserving deformation
J. reine angew. Math. 690 (2014), 134
The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones. Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painlevé equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a canonical Hamiltonian system whose Hamiltonian functions are polynomials in the canonical variables.

Projective reduction of the discrete Painlevé system of type A2 + A1 (with K. Kajiwara and N. Nakazono)
Int. Math. Res. Not. 2011 (2011), 930966
We consider the qPainlevé III equation arising from the birational representation of the affine Weyl group of type A2 + A1. We study the reduction of the qPainlevé III equation to the qPainlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the taufunctions.

From KP/UC hierarchies to Painlevé equations
Int. J. Math. 23 (2012), 1250010 (59pp)
We study the underlying relationship between Painlevé equations and infinitedimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painlevé equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g., Lax formalism, Hirota bilinear relations for taufunctions, Weyl group symmetry, and algebraic solutions in terms of the character polynomials, i.e., the Schur function and the universal character.

Constructing twodimensional integrable mappings that possess invariants of high degree (with J. Matsukidaira, A. Nobe and H. Tanaka)
RIMS Koukyuroku Bessatsu B13 (2009), 7584
We propose a method for constructing twodimensional integrable mappings that possess invariants with degree higher than two. Such integrable mappings are obtained by making a composition of a QRT mapping and a mapping that preserves the invariant curve of the QRT mapping except for changing the integration constant involved. We show several concrete examples whose integration constants change with period 2 and 3.

Ultradiscretization of a solvable twodimensional chaotic map assciated with the Hesse cubic curve (with K. Kajiwara, M. Kaneko and A. Nobe)
Kyushu J. Math. 63 (2009), 315338
We present a solvable twodimensional piecewise linear chaotic map that arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the nontrivial ultradiscrete limit of the solution in spite of a problem known as 'the minussign problem.'

On an integrable system of qdifference equations satisfied by the universal characters: its Lax formalism and an application to qPainlevé equations
Comm. Math. Phys. 293 (2010), 347359
The universal character is a generalization of the Schur function attached to a pair of partitions. We study an integrable system of qdifference equations satisfied by the universal characters, which is an extension of the qKP hierarchy and is called the lattice qUC hierarchy. We describe the lattice qUC hierarchy as a compatibility condition of its associated linear system (Lax formalism) and explore an application to the qPainlevé equations via similarity reduction. In particular a higherorder analogue of the qPainlevé VI equation is presented.

A geometric approach to taufunctions of difference Painlevé equations
Lett. Math. Phys. 85 (2008), 6578
We present a unified description of birational representation of Weyl groups associated with Tshaped Dynkin diagrams, by using a particular configuration of points in the projective plane. A geometric formulation of taufunctions is given in terms of defining polynomials of certain curves. If the Dynkin diagram is of affine type (E_6, E_7 or E_8), our representation gives rise to the difference Painlevé equations.

Ultradiscretization of solvable onedimensional chaotic maps (with K. Kajiwara and A. Nobe)
J. Phys. A: Math. Theor. 41 (2008), 395202 (13pp)
We consider the ultradiscretization of a solvable onedimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed.

A class of integrable and nonintegrable mappings and their dynamics (with B. Grammaticos, A. Ramani and T. Takenawa)
Lett. Math. Phys. 82 (2007), 3949
We analyse a class of mappings which by construction do not belong to the QRT family. We show that some of the members of this class have invariants of high degree. A new linearisable mapping is also identified. A mapping which possesses confined singularities while having nonzero algebraic entropy is presented. Its dynamics are studied in detail and shown to be related intimately to the Fibonacci recurrence.

Tropical representation of Weyl groups associated with certain rational varieties (with T. Takenawa)
Adv. Math. 221 (2009), 936954
Starting from certain rational varieties blownup from (P^1)^N,
we construct a tropical, i.e., subtractionfree birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. We develop an algebrogeometric framework of taufunctions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type,
our construction yields a class of (higher order) qdifference Painlevé equations
and its algebraic degree grows quadratically.

Tropical Weyl group action via point configurations and taufunctions of the qPainlevé equations
Lett. Math. Phys. 77 (2006), 2130
Starting from certain point sets in the projective plane, we construct a tropical (or subtractionfree birational) representation of Weyl groups over the field of taufunctions. In particular, our construction includes E_8, E_7, E_6 and D_5 as affine cases; each of them gives rise to the qdifference Painlevé equation.

Tau functions of qPainlevé III and IV equations
Lett. Math. Phys. 75 (2006), 3947
We present a geometric approach to taufunctions of the qPainlevé III and IV equations via rational surfaces with affine Weyl group symmetry of type A_2+A_1. We also study a similarity reduction of the qKP hierarchy to the equations.

Universal character and qdifference Painlevé equations
Math. Ann. 345 (2009), 395415
The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we introduce an integrable system of qdifference lattice equations satisfied by the universal character, and call it the lattice qUC hierarchy. We regard it as generalizing both qKP and qUC hierarchies. Suitable similarity and periodic reductions of the hierarchy yield the qdifference Painlevé equations of types A_{2g+1}, D_5, and E_6. As its consequence, a class of algebraic solutions of the qPainlevé equations is rapidly obtained by means of the universal character. In particular, we demonstrate explicitly the reduction procedure for the case of type E_6, via the framework of taufunctions based on the geometry of certain rational surfaces.

qPainlevé VI equation arising from qUC hierarchy (with T. Masuda)
Comm. Math. Phys. 262 (2006), 595609
We study the qdifference analogue of the sixth Painlevé equation (qPVI) by means of tau functions associated with affine Weyl group of type D5. We prove that a solution of qPVI coincides with a selfsimilar solution of the qUC hierarchy. As a consequence, we obtain in particular algebraic solutions of qPVI in terms of the universal character which is a generalization of Schur polynomial attached to a pair of partitions.

Universal characters and qPainlevé systems
Comm. Math. Phys. 260 (2005), 5973
We propose an integrable system of qdifference equations satisfied by the universal characters and regard it as a qanalogue of the UC hierarchy. Via a similarity reduction of this integrable system, rational solutions of the qPainlevé systems are constructed in terms of the universal characters.

Universal characters, integrable chains and the Painlevé equations
Adv. Math. 197 (2005), 587606
The universal character is a generalization of the Schur polynomial attached to a pair of partitions. We prove that the universal character solves the Darboux chain. The periodic closing of the chain is equivalent to the Painlevé equation of type A. Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.

Tau functions of the fourth Painlevé equation in two variables
Funkcial. Ekvac. 48 (2005), 137145
In this paper, we investigate the solutions of the fourth Painlevé equation in two variables by means of tau functions. Hirota bilinear forms and birational symmetries of the equation are presented. We obtain in particular a complete description of the rational solutions in terms of Schur polynomials.

Folding transformations of the Painlevé equations (with K. Okamoto and H. Sakai)
Math. Ann. 331 (2005), 713738
New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3 or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We make the complete list of such transformations up to birational symmetries. We also discuss correspondences of special solutions of Painlevé equations.

Integrable mappings via rational elliptic surfaces
J. Phys. A: Math. Gen. 37 (2004), 272130
We present a geometric description of the QRT map (which is an integrable mapping introduced by Quispel, Roberts, and Thompson) in terms of the addition formula of a rational elliptic surface. By this formulation, we classify all the cases when the QRT map is periodic; and show that its period is 2, 3, 4, 5 or 6. A generalization of the QRT map which acts birationally on a pencil of K3 surfaces, or CalabiYau manifolds, is also presented.

Rational solutions of the Garnier system in terms of Schur polynomials
Int. Math. Res. Not. 2003 (2003), 234158
We present a family of rational solutions of the Garnier system; and show that the corresponding tau functions form special polynomials under a certain normalization. We give an explicit formula for the special polynomials in terms of the Schur polynomial attached to a rectangular Young diagram.

Universal characters and an extension of the KP hierarchy
Comm. Math. Phys. 248 (2004), 501526
The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we define vertex operators which play roles of raising operators for the universal character. By means of the vertex operators, we obtain a series of nonlinear partial differential equations of infinite order, called the UC hierarchy; we regard it as an extension of the KP hierarchy.
We investigate also solutions of the UC hierarchy; we show that the totality of the space of solutions forms a direct product of two infinitedimensional Grassmann manifolds, and describe its infinitesimal transformations in terms of infinitedimensional Lie algebra.

Toda equation and special polynomials associated with the Garnier system
Adv. Math. 206 (2006), 657683
We prove that a certain sequence of tau functions of the Garnier system satisfies Toda equation. We construct a class of algebraic solutions of the system by the use of Toda equation; then show that the associated tau functions are expressed in terms of the universal character, which is a generalization of Schur polynomial attached to a pair of partitions.

Birational symmetries, Hirota bilinear forms and special solutions of the Garnier systems in 2variables
J. Math. Sci. Univ. Tokyo. 10 (2003), 355371
Hirota bilinear forms of the Garnier system in 2variables, G(1,1,1,1,1), are given. By using Hirota bilinear forms we construct new birational symmetries of G(1,1,1,1,1). We obtain special solutions of the Garnier system in nvariables, which are described in terms of solutions of the Garnier system in (n1)variables. We investigate also algebraic solutions for n=2.

Two approximation problems by Hermite, and Schlesinger transformations (in Japanese) (with T. Mano)
RIMS Kokyuroku Bessatsu B47 (2014), 7786
UC hierarchy, monodromy preserving deformation and hypergeometric function (in Japanese) ("Hook representation of the universal character" in Appendix C)
RIMS Kokyuroku 1765 (2011), 154167
UC hierarchy, monodromy preserving deformation and hypergeometric function (in Japanese)
Reports of RIAM Symposium 22AOS8 (2011), 4349
Symmetrization of qPainlevé equations (in Japanese) (with K. Kajiwara and N. Nakazono)
Reports of RIAM Symposium 20MES7 (2009), 2128
A geometric approach to tropical Weyl group actions and qPainlevé equations (with T. Takenawa)
Oberwolfach Reports 5 (2008), 25882590
Universal character and qdifference Painlevé equations (in Japanese)
RIMS Kokyuroku 1473 (2006), 114
Infinitedimensional integrable systems associated with universal characters, and Painlevé equations (in Japanese)
Reports of RIAM Symposium 16MES1 (2005), 128136
Integrable mappings via rational elliptic surfaces (in Japanese)
RIMS Kokyuroku 1400 (2004), 3138
Universal characters and an extension of the KP hierarchy (in Japanese)
RIMS Kokyuroku 1310 (2003), 5464
Toda equation and special polynomials associated with the Garnier system (in Japanese)
RIMS Kokyuroku 1296 (2002), 128136
Hierarchical structure and symmetries of the Garnier systems (in Japanese)
RIMS Kokyuroku 1203 (2001), 5770
Bilinear forms of the Garnier systems in two variables (in Japanese)
Rokko Lectures in Mathematics 7 (2000), 173182
Universal characters and Integrable systems
Ph.D. thesis, The University of Tokyo, 2003

