UC Berkeley

We give an explicit raising operator formula for the modified Macdonald polynomials H¯μ(X: q,t), which follows from our recent formula for ∇ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions H¯1,n(X: q,t) that we call n-Macdonald polynomials, which reduce to a scalar multiple of H¯μ(X: q,t) when n=1. We conjecture that the coefficients of 1,n-Macdonald polynomials in terms of Schur functions belong to N[q,t], generalizing Macdonald positivity.