On the summability and convergence of formal solutions of linear q-difference-differential equations with constant coefficients

We consider the Cauchy problem for homogeneous linear q-difference-differential equations with constant coefficients. We characterise convergent, k-summable and multisummable formal power series solutions in terms of analytic continuation properties and growth estimates of the Cauchy data. We also introduce and characterise sequences preserving summability, which make a very useful tool, especially in the context of moment differential equations.