33 citations · 339 across the 43 of their papers we have counts for
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On -Middle Convolution and -Hypergeometric Equations
Yumi Arai, Kouichi Takemura
The -middle convolution was introduced by Sakai and Yamaguchi. In this paper, we reformulate -integral transformations associated with the -middle convolution. In particul…
On symmetry of q-Painlevé equations and associated linear equations
Kouichi Takemura
We investigate the symmetry of the linear q-difference equations which are associated with some q-Painlevé equations. We apply it for adjustment of the expression of the time evolu…
-Middle Convolution and -Painlevé Equation
Shoko Sasaki, Shun Takagi, Kouichi Takemura
A -deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear -difference equation associated with the -Painlevé VI equation. The…
q-Heun equation and initial-value space of q-Painlevé equation
Shoko Sasaki, Shun Takagi, Kouichi Takemura
We show that the q-Heun equation and its variants appear in the linear q-difference equations associated to some q-Painlevé equations by considering the blow-up associated to their…
Variants of confluent q-hypergeometric equations
Ryuya Matsunawa, Tomoki Sato, Kouichi Takemura
Variants of the q-hypergeometric equation were introduced in our previous paper with Hatano. In this paper, we consider degenerations of the variant of the q-hypergeometric equatio…
Variants of -hypergeometric equation
Naoya Hatano, Ryuya Matsunawa, Tomoki Sato +1
We introduce two variants of -hypergeometric equation. We obtain several explicit solutions of variants of -hypergeometric equation. We show that a variant of -hypergeomet…
Heun's differential equation and its q-deformation
Kouichi Takemura
The -Heun equation is a -difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solu…
Ultradiscrete limit of the spectral polynomial of the -Heun equation
Kentaro Kojima, Tsukasa Sato, Kouichi Takemura
It is known that the q-Heun equation has polynomial-type solutions in some special cases, and the condition for the accessory parameter E is described by the roots of the spectral…
Polynomial solutions of -Heun equation and ultradiscrete limit
Kentaro Kojima, Tsukasa Sato, Kouichi Takemura
We study polynomial-type solutions of the -Heun equation, which is related with quasi-exact solvability. The condition that the -Heun equation has a non-zero polynomial-type…
On -deformations of the Heun equation
Kouichi Takemura
The -Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particul…
On two-parameter solutions of simultaneous ultradiscrete Painleve II equation with parity variables
Hikaru Igarashi, Kouichi Takemura
We introduce a simultaneous ultradiscrete Painleve II equation with parity variables, which is shown to be more suitable for studying two-parameter solutions than the single second…
Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin +1
We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lame polynomial, plays fundamental roles in…
On reducible monodromy representations of some generalized Lamé equation
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin +1
In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the…
New Airy-type solutions of the ultradiscrete Painleve II equation with parity variables
Hikaru Igarashi, Shin Isojima, Kouichi Takemura
The q-difference Painleve II equation admits special solutions written in terms of determinant whose entries are the general solution of the q-Airy equation. An ultradiscrete limit…
Ultradiscrete Painlevé VI with Parity Variables
Kouichi Takemura, Terumitsu Tsutsui
We introduce a ultradiscretization with parity variables of the -difference Painlevé VI system of equations. We show that ultradiscrete limit of Riccati-type solutions of -Pa…
Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial
Kouichi Takemura
We study Heun's differential equation in the case that one of the singularities is apparent. In particular we conjecture a relationship with generalized hypergeometric differential…
Integral transformation of Heun's equation and some applications
Kouichi Takemura
It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler…
Introduction to middle convolution for differential equations with irregular singularities
Kouichi Takemura
We introduce middle convolution for systems of linear differential equations with irregular singular points, and we presend a tentative definition of the index of rigidity for them…
Integral transformation and Darboux transformation
Kouichi Takemura
We review Darboux-Crum transformation of Heun's differential equation. By rewriting an integral transformation of Heun's differential equation into a form of elliptic functions, we…
Integral representation of solutions to Fuchsian system and Heun's equation
Kouichi Takemura
We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the…
Finite-gap potential, Heun's differential equation and WKB analysis
Kouichi Takemura
We review several results on the finite-gap potential and Heun's differential equation, and we discuss relationships among the finite-gap potential, the WKB analysis and Heun's dif…
The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations
Kouichi Takemura
We obtain isomonodromic transformations for Heun's equation by generalizing Darboux transformation, and we find pairs and triplets of Heun's equation which have the same monodromy…
The Hermite-Krichever ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials
Kouichi Takemura
Several results including integral representation of solutions and Hermite-Krichever Ansatz on Heun's equation are generalized to a certain class of Fuchsian differential equations…
Heun equation and Painlevé equation
Kouichi Takemura
We relate two parameter solutions of the sixth Painlevé equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential…