## 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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### References:

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