  We study the differential polynomial rings which are defined using the
special geometry of the moduli spaces of Calabi-Yau threefolds. The higher
genus topological string amplitudes are expressed as polynomials in the
generators of these rings, giving them a global description in the moduli
space. At particular loci, the amplitudes yield the generating functions of
Gromov-Witten invariants. We show that these rings are isomorphic to the rings
of quasi modular forms for threefolds with duality groups for which these are
known. For the other cases, they provide generalizations thereof. We
furthermore study an involution which acts on the quasi modular forms. We
interpret it as a duality which exchanges two distinguished expansion loci of
the topological string amplitudes in the moduli space. We construct these
special polynomial rings and match them with known quasi modular forms for
non-compact Calabi-Yau geometries and their mirrors including local P^2 and
local del Pezzo geometries with E_5,E_6,E_7 and E_8 type singularities. We
provide the analogous special polynomial ring for the quintic.
