  Utilizing spectral residues of parameterized, recursively defined sequences,
we develop a general method for generating identities of composition sums.
Specific results are obtained by focusing on coefficient sequences of solutions
of first and second order, ordinary, linear differential equations. Regarding
the first class, the corresponding identities amount to a proof of the
exponential formula of labelled counting. The identities in the second class
can be used to establish certain geometric properties of the simplex of
bounded, ordered, integer tuples. We present three theorems that support the
conclusion that the inner dimensions of such an order simplex are, in a certain
sense, more ample than the outer dimensions. As well, we give an algebraic
proof of a bijection between two families of subsets in the order simplex, and
inquire as to the possibility of establishing this bijection by combinatorial,
rather than by algebraic methods.
