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Confluence of \(q\)-difference to difference for a Fuchsian system. (Confluence \(q\)-différence vers différence pour un système Fuchsien.) (French. English summary) Zbl 1073.39013

It is a classical fact that the linear difference system \(X(x+h) = A_{h}(x)X(x)\) degenerates, when the step \(h\) goes to \(0\), into the differential system \(\widetilde{X}'(x) = \widetilde{A}(x) \widetilde{X}(x)\), where \(\widetilde{A}(x) = \lim_{h \to 0} \frac{A_{h}(x) - I_{n}}{h}\). This gives rise to the usual Euler method. A multiplicative version involves \(q\)-difference systems \(X(qx) = A_{q}(x) X(x)\) and degeneracy when \(q \to 1\), Then, one must take \(\widetilde{A}(x) = \lim_{q \to 1} \frac{A_{q}(x) - I_{n}}{(q-1) x}\). This gives rise to so-called “\(q\)-analogies”.
The analytical properties of this kind of degeneracy have been studied for \(q\)-difference systems by J. Sauloy [Ann. Inst. Fourier 50, 1021–1071 (2000; Zbl 0957.05012)], under the name of “confluence”. The case of difference systems seems much more difficult. As an intermediate step, the author of the paper under review studies the degeneracy of \(q\)-differences to differences. A. Duval [Ann. Fac. Sci. Toulouse 12, 335–374 (2003; Zbl 1096.33009)] had tackled the regular case.
Here, the previous results are extended to regular singular (or Fuchsian) systems. This requires the definition and thorough study of a new and very interesting class of special functions used to solve difference systems with constant coefficients.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
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