  In this paper we consider generalized moment functions of higher order. These
functions are closely related to the well-known functions of binomial type
which have been investigated on various abstract structures. In our former
paper we investigated the properties of generalized moment functions of higher
order on commutative groups. In particular, we proved the characterization of
generalized moment functions on a commutative group as the product of an
exponential and composition of multivariate Bell polynomial and a sequence
additive functions. In the present paper we continue the study of generalized
moment function sequences of higher order in the more abstract setting, namely
we consider functions defined on a hypergroup. We characterize these functions
on the polynomial hypergroup in one variable by means of partial derivatives of
a composition of polynomials generating the polynomial hypergroup and an
analytic function. As an example, we give an explicit formula for moment
generating functions of rank at most two on the Tchebyshev hypergroup.
