On parametric $$0-$$Gevrey asymptotic expansions in two levels for some linear partial $$q-$$difference-differential equations

Abstract A novel asymptotic representation of the analytic solutions to a family of singularly perturbed $$q-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>-</mml:mo> </mml:mrow> </mml:math> difference-differential equations in the complex domain is obtained. Such asymptotic relation shows two different levels associated to the vanishing rate of the domains of the coefficients in the formal asymptotic expansion. On the way, a novel version of a multilevel sequential Ramis-Sibuya type theorem is achieved.