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arXiv · 1401.4564

Building meromorphic solutions of $q$-difference equations using a Borel-Laplace summation

Thomas Dreyfus
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Abstract

After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace transformations of convenient orders and convenient direction in order to construct a solution of the same equation that is meromorphic on $\mathbb{C}^{*}$. We use this theorem to construct explicitly an invertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on $\mathbb{C}^{*}$. Moreover, when the system is put in the Birkhoff-Guenther normal form, we prove that the solutions we compute are exactly the same as the one constructed by Ramis, Sauloy and Zhang.

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Thomas Dreyfus. 2014-09-22. Building meromorphic solutions of $q$-difference equations using a Borel-Laplace summation. https://doi.org/10.1093/imrn%2Frnu137

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