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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.

MSC:

34M99 Ordinary differential equations in the complex domain
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