Orthogonality of Milne's polynomials and raising operators

Abstract

The preceding paper by Milne deals with a number of issues in the theory of symmetric polynomials which are of great present interest. In this writing we show that certain identities established in the paper are connected to several recent developments. In particular, we show a connection between some specializations of Milne's polynomials F λ ( Z, γ; q, t) with corresponding specializations of the polynomials P( X; q, t) recently introduced by Macdonald (1988). We shall also show that Milne's polynomials may be given a Vertex operator setting which extends that given by Jing to the Hall-Littlewood polynomials in a recent Yale doctoral thesis. We should note that the considerations that follow also contain a new proof of the basic duality result of Macdonald (1988) for the polynomials P( X; q, t) a more transparent way to deal with the ubiquitous raising operators of Littlewood.