BUILDING MEROMORPHIC SOLUTIONS OF q-DIFFERENCE EQUATIONS USING A BOREL-LAPLACE SUMMATION

Thomas Dreyfus (1)

1 IMJ - Institut de Mathématiques de Jussieu

After introducing q-analogs 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 C *. We use this theorem to construct explicitly an in-vertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on C *. Moreover, when the system has two slopes and is put in the Birkhoff-Guenther normal form, we show how the solutions we compute are related to the one constructed by Ramis, Sauloy and Zhang.

Domaines: Mathématiques générales [math.GM]

Thomas Dreyfus. BUILDING MEROMORPHIC SOLUTIONS OF q-DIFFERENCE EQUATIONS USING A BOREL-LAPLACE SUMMATION. International Mathematics Research Notices, 2015, 2015 (15), pp.6562 - 6587. ⟨10.1093/imrn/rnu137⟩. ⟨hal-01897301⟩