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Differential Galois groups and G-functions. (English) Zbl 0731.12004

Differential equations and computer algebra, Proc. Workshop, Ithaca/NY (USA) 1990, 149-180 (1991).
[For the entire collection see Zbl 0717.00014.]
Consider an n-dimensional system of first order linear differential equations with coefficients in the local Laurent series at the point 0, say. A theorem of Ramis then states that the differential Galois group is the Zariski closure in GL(n,\({\mathbb{C}})\) of the group generated by the formal monodromy, an exponential torus and the Stokes matrices, all relative to a given formal solution. In particular, when a system of first order equations over the rational functions has at most two singularities, at most one irregular, the local Galois group at the irregular singularity equals the global differential group. The author of the present paper applies this principle to some examples of confluent hypergeometric equations. The nicest example is the equation of order 7 having \(G_ 2\) as Galois group. This was also discovered by N. M. Katz. Fuller details on computations are to be found in A. Duval and the author [Pac. J. Math. 138, 25-56 (1989; Zbl 0705.34068)].

MSC:

12H20 Abstract differential equations
34G10 Linear differential equations in abstract spaces
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
34E05 Asymptotic expansions of solutions to ordinary differential equations