Multisummability and Stokes multipliers of linear meromorphic differential equations. (English) Zbl 0729.34005

Consider the differential equation \(xy'(x)=A(x)y(x)\) where \(y(x)\in {\mathbb{C}}^ n\) and \(A(x)\) is an analytic \((n\times n)\)-matrix having a pole at \(x=0\). It is well known that there exists a formal fundamental matrix \(\hat Y(x)=\hat F(t)t^{\Lambda}\exp Q(t^{-1}),\quad t=x^{1/p},\) where \(p\in {\mathbb{N}}\), Q(t) is a diagonal matrix whose entries are polynomials, \(\Lambda\) is a constant matrix satisfying \(\Lambda Q(t)=Q(t)\Lambda\), and \(\hat F(t)\) is an \((n\times n)\)-matrix whose entries are formal power series in t. In general, the series \(\hat F(t)\) does not converge. It is shown that \(\hat F(t)\) is multisummable in certain sectors such that the multisums \(F(t)\) satisfy \(F(t)\sim \hat F(t)\) as \(t\to 0\) and such that \(Y(x)=F(t)t^{\Lambda}\exp Q(t^{- 1}),\quad t=x^{1/p},\) is a fundamental matrix of the differential equation.


34M99 Ordinary differential equations in the complex domain
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[1] {\scW. Balser, B. L. J. Braaksma, J.-P. Ramis, and Y. Sibuya}, Multisummability of formal power series solutions of linear ordinary differential equations, Asympt. Anal., to appear. · Zbl 0754.34057
[2] Balser, W; Jurkat, W.B; Lutz, D.A, Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. math. anal. appl., 71, 48-94, (1979) · Zbl 0415.34008
[3] Balser, W; Jurkat, W.B; Lutz, D.A; 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, II, SIAM J. math. anal., SIAM J. math. anal., 19, 396-443, (1988) · Zbl 0639.34034
[4] Braaksma, B.L.J, Laplace integrals, factorial series and singular differential systems, (), 101-121, MCTract 100 · Zbl 0417.34013
[5] Braaksma, B.L.J, Laplace integrals in singular differential and difference equations, (), 25-53 · Zbl 0449.34005
[6] Braaksma, B.L.J; Harris, W.A, Laplace integrals and factorial series in singular functional differential systems, Appl. anal., 8, 23-45, (1978) · Zbl 0406.34058
[7] Ecalle, J; Ecalle, J, LES fonctions résurgentes, III, Publ. math. orsay, Publ. math. orsay, (1985) · Zbl 0602.30029
[8] Ecalle, J, L’accélération des fonctions résurgentes, (1987), manuscript
[9] {\scJ. Ecalle}, Calcul accélératoire et applications in “Travaux en Cours,”, Hermann, Paris, to appear.
[10] Harris, W.A; Sibuya, J.Y; Weinberg, L, Holomorphic solutions of linear differential systems at singular points, Arch. rational mech. anal., 35, 245-248, (1969) · Zbl 0227.34003
[11] Horn, J; Horn, J, Integration linearer differentialgleichungen durch laplacesche integrale und fakultätenreihen, Jahresber. Deutsch. math. ver., Jahresber. Deutsch. math. ver., 25, 74-83, (1917) · JFM 45.0487.01
[12] Immink, G.K, A note on the relation between Stokes multipliers and formal solutions of analytic differential equations, SIAM J. math. anal., 21, 782-792, (1990) · Zbl 0715.34011
[13] Jurkat, W.B, Meromorphe differentialgleichungen, () · Zbl 0408.34004
[14] {\scJ. Martinet and J.-P. Ramis}, Elementary acceleration and multisummability, Ann. Inst. H. Poincaré Phys. Théor., to appear. · Zbl 0748.12005
[15] Ramis, J.-P, LES séries k-sommables et leurs applications, (), 178-199 · Zbl 1251.32008
[16] Schäfke, R, Über das globale verhalten der normallösungen von X′(t) = (B + t−1A)X(t) und zweier arten von assozierten funktionen, Math. nachr., 121, 123-145, (1985) · Zbl 0563.34003
[17] Sibuya, Y, Linear differential equations in the complex domain: problems of analytic continuation, () · Zbl 0151.12503
[18] Turrittin, H.L, Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta math., 93, 27-66, (1955) · Zbl 0064.33603
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