## 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)

### Keywords:

linear difference system; $$q$$-difference systems

### Citations:

Zbl 0957.05012; Zbl 1096.33009
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