From Gauss to Painlevé : a modern theory of special functions : dedicated to Tosihusa Kimura

1. Elements of Differential Equations.- 1.1 Cauchy's existence theorem.- 1.2 Linear equations.- 1.3 Local behavior around regular singularities (Frobenius's method).- 1.4 Fuchsian equations.- 1.5 Pfaffian systems and integrability conditions.- 1.6 Hamiltonian systems.- 2. The Hypergeometric Differential Equation.- 2.1 Definition and basic facts.- 2.1.1 The Gauss hypergeometric equation.- 2.1.2 Hypergeometric series.- 2.1.3 Finite group action and Rummer's 24 solutions.- 2.2 Contiguity relations.- 2.2.1 Contiguity relations.- 2.2.2 Contiguity relations and particular solutions of the Toda equation.- 2.3 Integral representations.- 2.3.1 Integral representations as a tool for global problems.- 2.3.2 Euler integral representation derived from the power series.- 2.3.3 The Euler transform.- 2.3.4 The hypergeometric Euler transform.- 2.3.5 Barnes integral representation: interpolation method.- 2.3.6 Barnes integral representation: difference equation method.- 2.3.7 The Gauss-Kummer identity.- 2.4 Monodromy of the hypergeometrie equation.- 2.4.1 Fundamental group of ?1 \ {0, 1, ?} and the monodromy of the Riemann equation.- 2.4.2 Classification of 2-dimensional representations of the free group with 2 generators.- 2.4.3 Finding the monodromy by local properties and the Fuchs relation.- 2.4.4 Finding the monodromy by Euler integrals over arcs.- 2.4.5 Finding the monodromy by Euler integrals over double loops.- 2.4.6 Finding the monodromy by Barnes integrals.- 2.4.7 Finding the monodromy by Gauss-Kummer's identity.- 3. Monodromy-Preserving Deformation, Painleve Equations and Garnier Systems.- 3.1 Painleve equations.- 3.1.1 Historical remarks.- 3.1.2 Relations between the PJ's.- 3.1.3 Symmetry of the Painleve equation PVI.- 3.1.4 Solutions of PVI at singular points.- 3.1.5 Hamiltonian structure for PVI.- 3.1.6 Particular solutions of the PJ's.- 3.2 The Riemann-Hilbert problem for second order linear differential equations.- 3.2.1 Spaces of Fuchsian differential equations and those of representations of ?1.- 3.2.2 The Riemann-Hilbert problem.- 3.3 Monodromy-preserving deformations.- 3.3.1 M-invariant fundamental solutions.- 3.3.2 Totality of M-invariant fundamental solutions.- 3.3.3 Monodromy-preserving deformation of second order differential equations.- 3.3.4 SL-equations.- 3.3.5 Deformation equations for second order SL-equations.- 3.4 The Garnier system n.- 3.4.1 Main theorem.- 3.4.2 Reduction to SL-equations.- 3.4.3 Explicit forms of Ki and Li.- 3.4.4 Explicit expression of Ai(x, t).- 3.4.5 Proof of Theorem 4.2.2.- 3.5 Schlesinger systems.- 3.6 The Schlesinger system and the Garnier system n.- 3.6.1 Transformation of systems of equations into second order equations.- 3.6.2 Transformation of the Schlesinger system into the Garnier system.- 3.6.3 Transformation of second order equations into systems of equations.- 3.6.4 Relation between the Garnier system and the Schlesinger system.- 3.7 The polynomial Hamiltonian system ?nassociated with n.- 3.7.1 Transformation of ninto the polynomial Hamiltonian system ?n.- 3.7.2 Proof of Theorem 7.1.2.- 3.7.3 r-function associated with ?n.- 3.8 Symmetries of the Garnier system nand of the system ?n.- 3.8.1 Symmetries of n.- 3.8.2 Symmetries of ?n.- 3.8.3 Prolongation of the system ?n.- 3.9 Particular solutions of the Hamiltonian system ?n.- 3.9.1 The Lauricella hypergeometric series FD.- 3.9.2 Particular solutions of ?n.- 4. Studies on Singularities of Non-linear Differential Equations.- 4.1 Singularities of regular type.- 4.1.1 Holomorphic solutions.- 4.1.2 One-dimensional case.- 4.1.2.1 Formal transformations.- 4.1.2.2 Convergence of formal transformations.- 4.1.3 The n-dimensional case.- 4.2 Fixed singular points of regular type of Painleve equations.- 4.2.1 Transformation into the normal form.- 4.2.2 Solutions of equations in normal form.- 4.2.3 Proof of Theorem 2.2.1.- 4.2.4 Solutions of Painleve equations.- Notes on the chapter titlepage illustrations.- Index of symbols.