Some new symmetric function tools and their applications

We prove a technical identity involving the Δ operator from Macdonald polynomial theory, which allows us to transform expressions involving the Δ 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 occur-ing 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 Δ operator and the power sum symmetric function p n .