MR Davide GUZZETTI Davide GUZZETTI On Stokes Matrices in terms of Connection Coefficients Funkcialaj Ekvacioj. Serio Internacia 59 2016 383--433 http://fe.math.kobe-u.ac.jp/FE/FullPapers/59-3/59_383.pdf http://www.ams.org/mathscinet-getitem?mr=MR We study the classical problem of the computation of a complete system of Stokes matrices in terms of connection coefficients. Stokes matrices refer to a linear system of ODEs with Poincar\'e rank one and semi-simple leading matrix, while the connection coefficients connect solutions of the associated hypergeometric system of ODEs. The problem here is solved with no assumptions on the residue matrix at zero of the system of Poincar\'e rank one, so extending method and results of [4]. Stokes matrices, Connection matrices, Monodromy linear differential systems, Laplace transform. 34M40, 34M35. 59-383 2016 On Stokes Matrices in terms of Connection Coefficients Davide GUZZETTI Davide GUZZETTI
1 Balser, W. Birkhoff's reduction problem Russian Math. Surveys 59 2004 1047-1059 MR2138466
2 Balser, W.; Jurkat, W. B.; Lutz, D. A. Birkhoff Invariants and Stokes' Multipliers for Meromorphic Linear Differential Equations J. Math. Anal. Appl. 71 1979 48-94 MR0545861
3 Balser, W.; Jurkat, W. B.; Lutz, D. A. A General Theory of Invariants for Meromorphic Differential Equations; Part II, Proper Invariants Funkcial. Ekvac. 22 1979 257-283 MR0577846 0577846
4 Balser, W.; Jurkat, W. B.; Lutz, D. A. On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities, I SIAM J. Math. Anal. 12 1981 691-721 MR0625827
5 Balser, W.; Jurkat, W. B.; Lutz, D. A. Transfer of Connection Problem for Meromorphic Differential Equations of Rank $r\geq 2$ and Representation of Solutions J. Math. Anal. Appl. 85 1982 488-542 MR0649188
6 Birkhoff, G. D. Singular points of ordinary linear differential equations Trans. Amer. Math. Soc. 10 1909 436-470 MR1500848
7 Doetsch, G. Introduction to the Theory and Application of the Laplace Transformation Springer-Verlag, New York-Heidelberg 1974 MR0344810
8 Dubrovin, B. Geometry of 2D topological field theories Lecture Notes in Math., 1620, Springer, Berlin 1996 pp. 120-348 MR1397274
9 Dubrovin, B. Painlevé trascendents in two-dimensional topological field theory CRM Ser. Math. Phys., Springer, New York 1999 pp. 287-412 MR1713580
10 Dubrovin, B. On almost duality for Frobenius manifolds Amer. Math. Soc. Transl. Ser. 2, 212, Amer. Math. Soc., Providence, RI 2004 pp. 75-132 MR2070050
11 Guzzetti, D., arXiv:1407.1206 (2014) 12 Écalle, J. Les Fonctions Résurgentes, I Publications Mathématiques d'Orsay 81, 5, Université de Paris-Sud, Département de Mathématique, Orsay 1981 MR0670417 13 Écalle, J. Les Fonctions Résurgentes, II Publications Mathématiques d'Orsay 81, 6, Université de Paris-Sud, Département de Mathématique, Orsay 1981 MR0670418 14 Écalle, J. Les Fonctions Résurgentes, III Publications Mathématiques d'Orsay 85, 5, Université de Paris-Sud, Département de Mathématique, Orsay 1985 MR0852210
15 Galkin, S.; Golyshev, V.; Iritani, H. Gamma Classes and Quantum Cohomology of Fano Manifolds: Gamma Conjectures Duke Math. J. 165 2016 2005-2077 MR3536989
16 Ince, E. L. Ordinary differential equations Dover Publications, New York 1956 MR0010757
17 Loday-Richaud, M. Rank Reduction, Normal Forms and Stokes Matrices Expo. Math. 19 2001 229-250 MR1852074
18 Loday-Richaud, M.; Remy, P. Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems J. Differential Equations 250 2011 1591-1630 MR2737219
19 Lutz, D. A. On the reduction of rank of linear differential systems Pacific J. Math. 42 1972 153-164 MR0318551
20 Mazzocco, M. Painlevé sixth equation as isomonodromic deformations equation of an irregular system CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence, RI 2002 pp. 219–238 MR1916784
21 Remy, P. Matrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique Ann. Fac. Sci. Toulouse Math. (6) 21 2012 93-150 MR2954106
22 Turrittin, H. L. Reducing the rank of ordinary differential equations Duke Math. J. 30 1963 271-274 MR0148967
23 Ugaglia, M. On a Poisson Structure on the Space of Stokes Matrices Internat. Math. Res. Notices 1999 1999 473-493 MR1692595
24 Wasow, W. Asymptotic Expansions for Ordinary Differential Equations Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1965 MR0203188