On k-summability of formal solutions to the Cauchy problems for some linear q-difference-differential equations

摘要

We consider the Cauchy problem for linear q-difference-differential equations which are q-analogs of linear partial differential equations. In this paper, we give a necessary and sufficient condition for the k-summability of their formal solutions, though they were treated in the category of the theory of q-Borel summability in many previous works. Moreover, we see that the condition is different from a q-analog of the one for corresponding partial differential equations.