Systèmes aux \(q\)-différences singuliers réguliers : classification, matrice de connexion et monodromie. (Regular singular \(q\)-difference systems: Classification, connection matrix and monodromy). (French) Zbl 0957.05012

This paper is connected with the linear \(q\)-difference system \(\sigma_{q}X=AX\), \(A \in \text{GL}_{n}(C(z))\), with rational coefficients. Local solutions are constructed in the case of some unresounding system with \(A(0)\in\text{GL}_{n}(C)\) and in the case of a system with constant coefficients. A global characterization of regular singular systems is given. Next the author studies the behavior for \(q \rightarrow 0\) of the canonical solutions and for the solutions with a fixed initial value. The behavior of the connection matrix for \(q \rightarrow 0\) is also investigated. Finally, three examples are given.


05A30 \(q\)-calculus and related topics
39A13 Difference equations, scaling (\(q\)-differences)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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