  It is known that the defining relations of the orthosymplectic Lie
superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n
pairs of paraboson operators $b^\pm_i$. In particular, with the usual star
conditions, this implies that the ``parabosons of order p'' correspond to a
unitary irreducible (infinite-dimensional) lowest weight representation V(p) of
osp(1|2n). Apart from the simple cases p=1 or n=1, these representations had
never been constructed due to computational difficulties, despite their
importance. In the present paper we give an explicit and elegant construction
of these representations V(p), and we present explicit actions or matrix
elements of the osp(1|2n) generators. The orthogonal basis vectors of V(p) are
written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of
osp(1|2n) plays a crucial role. Our results also lead to character formulas for
these infinite-dimensional osp(1|2n) representations. Furthermore, by
considering the branching $ osp(1|2n) \supset sp(2n) \supset u(n)$, we find
explicit infinite-dimensional unitary irreducible lowest weight representations
of sp(2n) and their characters.
