  We first describe a canonical mirror partner (B-model) of the small quantum
orbifold cohomology of weighted projective spaces (A-model) in the framework of
differential equations: we attach to the A-model (resp. B-model) a D-module on
the torus and we show that these two D-modules are isomorphic. This makes the A
and B-models mirror partners and yields, in this situation, an explicit and
finer version of a recent result of Iritani. Then we study, using the theory of
the Kashiwara-Malgrange filtration, their degenerations at the origin and we
apply our results to the construction of (classical, limit, logarithmic)
Frobenius manifolds.
