A note on $G_q$-summability of formal solutions of some linear $q$-difference-differential equations (New development of microlocal analysis and singular perturbation theory)

Abstract

Let q > 1 and delta > 0 . For a function f(t, z) , the q-shift operator sigma_{q} in t is defined by sigma_{q}(f)(t, z) = f(qt, z). This article discusses a linear q-difference-differential equation sum_{j+delta|alpha|leq m}a_{j, alpha}(t, z)(sigma_{q})^{j}partial_{z} ^{alpha}X = F(t, z) in the complex domain, and shows a result on the Gq-summability of formal solutions (which may be divergent) in the framework of q-Laplace and q-Borel transforms by Ramis-Zhang.