q-ANALOGUE OF SUMMABILITY OF FORMAL SOLUTIONS OF SOME LINEAR q-DIFFERENCE-DIFFERENTIAL EQUATIONS

Let q > 1. The paper considers a linear q-difference-differential equation: it is a q-difference equation in the time variable t, and a partial differential equation in the space variable z. Under suitable conditions and by using q-Borel and q-Laplace transforms (introduced by J.-P. Ramis and C. Zhang), the authors show that if it has a formal power series solution (X) over cap (t; z) one can construct an actual holomorphic solution which admits (X) over cap (t; z) as a q-Gevrey asymptotic expansion of order 1.