Some New Symmetric Function Tools And Their Applications

We prove a technical identity involving the Delta operator from Macdonald polynomial theory, which allows us to transform expressions involving the Delta operator and the Hall scalar product into other such expressions. We show how our technical identity, although following easily from the well-known Koornwinder-Macdonald reciprocity theorem, contains as special cases several identities occuring in the literature, proved there by more complicated arguments. We also show how our identity can be used to obtain some new expressions for the q, t-Narayana numbers introduced by Dukes and Le Borgne, as well as new identities involving the Delta operator and the power sum symmetric function p(n).