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Article Dans Une Revue Mathematische Zeitschrift Année : 2021

Functional relations of solutions of $q$-difference equations

Julien Roques

Résumé

In this paper, we study the algebraic relations satisfied by the solutions of $q$-difference equations and their transforms with respect to an auxiliary operator. Our main tool is the parametrized Galois theories developed in two papers. The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals a $\mathbf{q'}$-difference operator. In both cases, we give criteria to guaranty the algebraic independence of a series, solution of a $q$-difference equation, with either its successive derivatives or its $\mathbf{q'}$-transforms. We apply our results to $q$-hypergeometric series.
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Dates et versions

hal-01959032 , version 1 (28-09-2021)

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Thomas Dreyfus, Charlotte Hardouin, Julien Roques. Functional relations of solutions of $q$-difference equations. Mathematische Zeitschrift, 2021, ⟨10.1007/s00209-020-02669-4⟩. ⟨hal-01959032⟩
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