  The notion of $F$-signature is defined by C. Huneke and G. Leuschke and this
numerical invariant characterizes some singularities. This notion is extended
to finitely generated modules and called dual $F$-signature. In this paper, we
determine the dual $F$-signature of a certain class of Cohen-Macaulay modules
(so-called "special") over cyclic quotient surface singularities. Also, we
compare the dual $F$-signature of a special Cohen-Macaulay module with that of
its Auslander-Reiten translation. This gives a new characterization of the
Gorensteiness.
