On singularly perturbed <Emphasis Type="Italic">q</Emphasis>-difference-differential equations with irregular singularity

We study a q-analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain. We construct solutions of this problem that are holomorphic on open half-q-spirals. Using a version of a q-analog of the Malgrange–Sibuya theorem obtained by J.-P. Ramis, J. Sauloy, and C. Zhang, we show the existence of a formal power-series solution in the perturbation parameter which is the q-asymptotic expansion of these holomorphic solutions.