Hermite-Pade approximation, isomonodromic deformation and hypergeometric integral
Hermite-Pade approximation, isomonodromic deformation and hypergeometric integral
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Hermite-Pade 近似、等单向变形和超几何积分
DOI：
10.1007/s00209-016-1713-y
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发表时间：
2017
影响因子：
0.8
通讯作者：
Toshiyuki Mano and Teruhisa Tsuda
中科院分区：
文献类型：
--
作者：
Yuki Fuseya;Masao Ogata;Hidetoshi Fukuyama;伏屋雄紀，小形正男，福山秀敏;小形正男，伏屋雄紀，福山秀敏;江本裕行，安藤裕一郎，仕幸英治，伏屋雄紀，新庄輝也，白石誠司;Yuki Fuseya;伏屋雄紀;伏屋雄紀;伏屋雄紀，小形正男，福山秀敏;冠木悠太郎，三本啓輔，赤津光洋，根本祐一，後藤輝孝，伏屋雄紀;Toshiyuki Mano and Teruhisa Tsuda
我们发展了有理逼近理论和等单行变形理论之间的内在联系。我们证明了Hermite的两个函数逼近问题中的某种对偶性导致了Schlesinger变换，即线性微分方程的变换使其特征指数移位整数，而保持其单调不变。由于近似式和余式是用块Toeplitz行列式来描述的，所以人们可以清楚地理解等序形变中的行列式结构。我们在包括第六类Painlevé方程和Garnier系统在内的一类等偏型哈密顿系统中证明了我们的方法，特别地，我们用迭代超几何积分的形式给出了它们的解。文中还讨论了利用向量连分式构造Schlesinger变换的算法。
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 block-Toeplitz 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.