2017 Volume 60 Issue 1 Pages 21-63
We study an inhomogeneous linear q-difference differential Cauchy problem, with a complex perturbation parameter ε, whose coefficients depend holomorphically on ε and on time in the vicinity of the origin in C2 and are bounded analytic on some horizontal strip in C w.r.t the space variable. This problem is seen as a q-analog of an initial value problem recently investigated by the author and A. Lastra in [9]. Here a comparable result with the one in [9] is achieved, namely we construct a finite set of holomorphic solutions on a common bounded open sector in time at the origin, on the given strip above in space, when ε belongs to a well selected set of open bounded sectors whose union covers a neighborhood of 0 in C*. These solutions are constructed through a continuous version of a q-Laplace transform of some order k ≥ 1 introduced newly in [6] and Fourier inverse map of some function with q-exponential growth of order k on adequate unbounded sectors in C and with exponential decay in the Fourier variable. Moreover, by means of a q-analog of the classical Ramis-Sibuya theorem, we prove that they share a common formal power series (that generally diverge) in ε as q-Gevrey asymptotic expansion of order 1/k.
