  Sums of the form $\sum_{q \leq N_1 < \cdots < N_m \leq n}{a_{(m);N_m}\cdots
a_{(2);N_2}a_{(1);N_1}}$ date back to the sixteen century when Vi\`ete
illustrated that the relation linking the roots and coefficients of a
polynomial had this form. In more recent years, such sums have become
increasingly used with a diversity of applications. In this paper, we develop
formulae to help with manipulating such sums (which we will refer to as
multiple sums). We develop variation formulae that express the variation of
multiple sums in terms of lower order multiple sums. Additionally, we derive a
set of partition identities that we use to prove a reduction theorem that
expresses multiple sums as a combination of simple (non-recurrent) sums. We
present a variety of applications including applications concerning polynomials
and MZVs as well as a generalization of the binomial theorem. Finally, we
establish the connection between multiple sums and a type of sums called
recurrent sums. By exploiting this connection, we provide additional partition
identities for odd and even partitions.
