Some theorems and conjectures for the Dyson polynomials
The Macdonald polynomials, which extend the Schur functions, satisfy many algebraic and combinatorial properties which extend those of the Schur functions. They also satisfy many properties which are more analytic and are related to Selberg's integral, an equivalent constant term identity due to Morris, and certain constant term identities associated with root systems. These intersect in an orthogonality relation which can be expressed in terms of two inner products.
The Dyson polynomials extend the Macdonald polynomials to many parameters.
We give some theorems and conjectures. Using Good's proof, we give a constant
term orthogonality relation for partitions with one non zero part.