Multisummability of formal power series solutions of linear ordinary differential equations. (English) Zbl 0754.34057

Balser, W.; Braaksma, B. L. J.; Ramis, Jean-Pierre; Sibuya, Yasutaka

Asymptotic Anal. 5, No. 1, 27-45 (1991).

The authors consider the differential equation \({\mathcal L}x=0\) when \({\mathcal L}=dx/dz-A(z)\), \(x\) is an \(n\times n\) matrix, \(z\) is a complex independently variable and \(A\) is an \(n\times n\) matrix of functions meromorphic at \(z=0\), and prove these two theorems.
Theorem 4.1. Let \({1\over 2}\leq h_ q<h_{q-1}<\cdots<h_ 2<h_ 1<+\infty\) be all the positive slopes of the Newton polygon \(N({\mathcal L})\) of the operator \({\mathcal L}\). If a formal power series \(f\in\mathbb{C}[[z]]^ n\) satisfies the condition \({\mathcal L}[f]\in(\mathbb{C}\{z\}[z^{-1}])^ n\) and if a direction \(\arg z=d\) is not singular, then there exist \(f_ \ell\in(\mathbb{C}\{z\}_{h_ \ell d})^ n\) \((\ell=1,2,\ldots,q)\) such that \(f=\sum_{q\geq\ell\geq 1}f_{\ell}\), \({\mathcal L}[f_ \ell]\in(\mathbb{C}\{z\}[z^{-1}])^ n\) \((\ell=1,2,\ldots,q)\). In the case when \(0<h_ \ell<{1\over 2}\) for some \(\ell\), we change the independent variable \(z\) by \(z=\zeta^ m\) so that \(mh_ \ell\geq{1\over 2}\) for all \(\ell\).
Theorem 6.1. For any given direction \(\arg z=d\), there exists an admissible system \(\{X_ \nu(z)\mid\nu\in\mathbb{Z}\}\) of fundamental solutions of equations (5.5) such that the corresponding Stokes multipliers \(\{V_ \nu\mid\nu\in\mathbb{Z}\}\) satisfy the condition \(V_ \nu^{(j-1)}+{\mathbf I}\) for every pair \((j,\nu)\) such that \(d-\pi/(2k_ j)<\tau_{\nu-1}<\tau_ \nu<d+\pi/(2k_ j)\), where \(j=1,2,\ldots,p\).

Reviewer: H.S.Nur (Fresno)

MSC:

34E15	Singular perturbations for ordinary differential equations
34M99	Ordinary differential equations in the complex domain
30B10	Power series (including lacunary series) in one complex variable
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A30	Linear ordinary differential equations and systems

Keywords:

multisummability of formal power series solutions; linear; differential equation; complex; meromorphic; formal power series; fundamental solutions; Stokes multipliers