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.