Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

Sakai, Harutaka

doi:10.1007/s002200100446

Cited by 524 publications

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“…In [111], Sakai gave a classification of discrete Painlevé equations connected with the affine Weyl groups. On the top of this scheme one has the elliptic Painlevé equation related to the root systemÊ 8 .…”

Section: Resultsmentioning

confidence: 99%

Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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Abstract. We give a brief review of the main results of the theory of elliptic hypergeometric functions -a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

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Section: Resultsmentioning

confidence: 99%

Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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“…In [4], it has been shown that the q-P IV coincides with Sakai's Mul.6 system [13]. As mentioned in Section 1, the q-P V (1.1) has W (A (1) 1 × A (1) 3 ) symmetry by the original construction.…”

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confidence: 94%

“…As mentioned in Section 1, the q-P V (1.1) has W (A (1) 1 × A (1) 3 ) symmetry by the original construction. On the other hand, Sakai's Mul.5 system [13], which should be also regarded as a q-analogue of the Painlevé V equation, admits the symmetry of W (A (1) 4 ). It might be an important problem to study the relationship between the equation (1.1) and Sakai's Mul.5 system.…”

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confidence: 99%

On the rational solutions of q-Painlevé V equation

Masuda¹

2003

Nagoya Mathematical Journal

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“…In Sakai's theory [21] the discrete Painlevé equations were classified on the basis of rational surfaces connected to extended affine Weyl groups. There exist three types of discrete Painlevé equations in the classification: elliptic difference (e-), multiplicative difference (q-) and additive difference (d-).…”

Section: Introductionmentioning

confidence: 99%

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

Nagao

2016

Lett Math Phys

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Abstract. Recently we studied Padé interpolation problems of q-grid, related to q-Painlevé equations of type E (1) 7 , E (1) 6 , D (1) 5 , A (1) 4 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 (1) 5 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 (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) . Moreover determinant formulae of the special solutions for q-Painlevé equation of type E (1) 6 are given in terms of the terminating q-Appell Lauricella function. IntroductionIn this paper we continue [13] and apply the Padé approximation method to the q-Painlevé equations of type E (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) .1.1. The background of discrete Painlevé equations.In Sakai's theory [21] the discrete Painlevé equations were classified on the basis of rational surfaces connected to extended affine Weyl groups. There exist three types of discrete Painlevé equations in the classification: elliptic difference (e-), multiplicative difference (q-) and additive difference (d-). The discrete Painlevé equations of q-difference type are classified as follows:Here A → B means that B is obtained from A by degeneration. method to continuous Painlevé equations of type P VI , P V , P IV and Garnier system by using differential grid (i.e. Padé approximation).The Padé method for discrete Painlevé equations has been applied to the following types:( Remark 1. On the key points of the Padé methodThere are two key points to apply the Padé approximation/interpolation method [4,13,17,33]. The first key point is the appropriate choice of approximated/interpolated functions (see Table (2.2) and Remark 4). The second key point is to consider two linear q-difference three term relations (2.6) satisfied by the error terms of the Padé approximation/interpolation problems. Then the error terms can be expressed in terms of special solutions of q-Painlevé equations. Therefore the q-difference relations are the main subject in our study, and they naturally give the evolution equations, the Lax pairs and the special solutions for the corresponding q-Painlevé equations. Remark 2. On a connection between the Padé method and the theory of semiclassical orthogonal polynomialsThe connection between semiclassical orthogonal polynomials (classical orthogonal polynomials related to a suitable weight function) and Painlevé/Garnier systems has been demonstrated in [11]. It has been shown that coefficients of three term recurrence relations, satisfied by several semiclassical orthogonal polynomials, can be expressed in terms of sol...

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Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

Cited by 524 publications

References 14 publications

Essays on the theory of elliptic hypergeometric functions

Essays on the theory of elliptic hypergeometric functions

On the rational solutions of q-Painlevé V equation

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

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