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.
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