Pfaffian Formulation of Schur's Q-functions

We introduce a Pfaffian formula that extends Schur's Q-functions Q_ to be indexed by compositions with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing Q_ into sums indexed by partitions with removed parts. Consequently, we are able to prove several identities of Schur's Q-functions using only simple algebraic methods.