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

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 ${\mathbb{C}}^{\ast }$. 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 ${\mathbb{C}}^{\ast }$. 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.

To appear in International Mathematics Research Notices. IMRN

References in corpus (2)

• Algebraic construction of the Stokes sheaf for irregular linear q-difference equations
• The q-analogue of the wild fundamental group (II)

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• The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_r{\phi }_s(a;b;q,x)$
• Isomonodromic deformation of q-difference equations and confluence
• On parametric multilevel q-Gevrey asymptotics for some linear Cauchy problem
• On the summability of formal solutions of some linear q-difference-differential equations
• On integral representations of $q$-difference operators and their applications