Differential Galois groups of confluent generalized hypergeometric equations: An approach using Stokes multipliers. (English) Zbl 0883.12004

The aim of this paper is to “analytically” compute the (global) differential Galois group of confluent generalized hypergeometric equations. This differential equation has two singularities, one of which is regular. Then the global differential Galois group is topologically generated by the two local Galois groups near singularities. It is a classical result that, near the regular singularity, the local Galois group is topologically generated by the monodromy. Near the irregular singularity, Ramis showed the local Galois group to be topologically generated by the “formal monodromy”, the so-called “Stokes matrices” and the “exponential torus”. A concise and pleasant survey on all this theory is given in the first section.
After some generalities about confluent generalized hypergeometric equations \[ D_{pq}= (-1)^{q-p}z\prod^{p}_{j=1} (\partial+\mu_{i}) -\prod^{q}_{j=1}(\partial+\nu_{i}-1),\quad \partial=z {d \over dz} \] the Galois group (that depends on parameters) is explicitly computed for \( D_{42}\), \( D_{51}\), \( D_{71} \) and, under the hypothesis of self-duality, for \( D_{2q,2q-2}\). Even if rather computational, this paper is a beautiful illustration of the powerfulness of analytical methods.
Reviewer: G.Christol (Paris)


12H05 Differential algebra
33C20 Generalized hypergeometric series, \({}_pF_q\)
34A30 Linear ordinary differential equations and systems
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