  In this paper we continue the discussion about relations between exponential
polynomials and generalized moment generating functions on a commutative
hypergroup. We are interested in the following problem: is it true that every
finite dimensional variety is spanned by moment functions? Let $m$ be an
exponential on $X$. In our former paper we have proved that if the linear space
of all $m$-sine functions in the variety of an $m$-exponential monomial is (at
most) one dimensional, then this variety is spanned by moment functions
generated by $m$. In this paper we show that this may happen also in cases
where the $m$-sine functions span a more than one dimensional subspace in the
variety. We recall the notion of a polynomial hypergroup in $d$ variables,
describe exponentials on it and give the characterization of the so called
$m$-sine functions. Next we show that the Fourier algebra of a polynomial
hypergroup in $d$ variables is the polynomial ring in $d$ variables. Finally,
using Ehrenpreis--Palamodov Theorem we show that every exponential polynomial
on the polynomial hypergroup in $d$ variables is a linear combination of moment
functions contained in its variety.
