q-analogue of summability of formal solutions of some linear q-difference-differential equations

Abstract

Let \(q\gt 1\). The paper considers a linear \(q\)-difference-differential equation: it is a \(q\)-difference equation in the time variable \(t\), and a partial differential equation in the space variable \(z\). Under suitable conditions and by using \(q\)-Borel and \(q\)-Laplace transforms (introduced by J.-P. Ramis and C. Zhang), the authors show that if it has a formal power series solution \(\hat{X}(t,z)\) one can construct an actual holomorphic solution which admits \(\hat{X}(t,z)\) as a \(q\)-Gevrey asymptotic expansion of order \(1\).

Keywords

• \(q\)-difference-differential equations
• summability
• formal power series solutions
• \(q\)-Gevrey asymptotic expansions
• \(q\)-Laplace transform