  We calculate the period integrals for a special class of affine hypersurfaces
(deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their
Mellin transforms. A description of the relation between poles of Mellin
transforms of period integrals and the mixed Hodge structure of the cohomology
of the hypersurface is given. By interpreting the period integrals as solutions
to Pochhammer hypergeometric differential equation, we calculate concretely the
irreducible monodromy group of period integrals that correspond to the
compactification of the affine hypersurface in a complete simplicial toric
variety. As an application of the equivalence between oscillating integral for
Delsarte polynomial and quantum cohomology of a weighted projective space
$\mathbb{P}_{\bf B}$, we establish an equality between its Stokes matrix and
the Gram matrix of the full exceptional collection on $\mathbb{P}_{\bf B}$.
