Transcendence of solutions of q-Painlevé equation of type $${A_7^{(1)}}$$

Nishioka, Seiji

doi:10.1007/s00010-010-0007-4

Cited by 16 publications

(16 citation statements)

References 7 publications

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“…The Painlevé equations are believed to define new transcendental functions, and it was rigorously proved for the Painlevé differential equations (see, for example [128,129]). Similar investigations for discrete Painlevé equations have been done in [80]. Asymptotic analysis for the solutions of Painlevé equations are also an important subject in view of applications [10,13,14,42,64].…”

Section: Introductionmentioning

confidence: 72%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

124

250

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

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Section: Introductionmentioning

confidence: 72%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

124

250

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“…Lemma 10 (Lemma 2 in S. Nishioka [10]). Let t be transcendental over C, F =CðtÞ a finite algebraic field extension, and t A AutðF =CÞ satisfy tt ¼ t þ 1.…”

Section: Proof Of Irreducibilitymentioning

confidence: 99%

Irreducibility of Discrete Painlevé Equation of Type D7(1)

Nishioka

2017

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Abstract. In this paper, we will study the irreducibility of the discrete Painlevé equation of type D ð1Þ 7 in the sense of decomposable extensions. The irreducibility here particularly implies that the transcendental function solution cannot be built from rational functions by reiterating algebraic operations, the taking of a solution of a linear di¤erence equation and the taking of a solution of a first-order algebraic di¤erence equation. We also study non-existence of algebraic function solution. A modification to the definition of the decomposable extension is mentioned.

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“…(Proposition in [6]) Let q ∈ C × be not a root of unity. Let f be a solution of q-P (A 6 ) a with a ∈ C × .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In [6] the author proved Proposition 4 which dealt with algebraic solutions of the q-Painlevé equation of type A (1) 6 ,…”

Section: Introductionmentioning

confidence: 99%