c-functions and Macdonald polynomials

TL;DR

This paper explores $c$-functions and Macdonald polynomials, providing new proofs and extending classical formulas, including the Boson-Fermion correspondence and Weyl character formula within this framework.

Contribution

It offers alternative proofs for key Macdonald polynomial formulas and establishes the Boson-Fermion correspondence and Weyl character formula in this context.

Findings

Derived $c$-function formulas for Macdonald polynomials

Provided alternative proofs for Macdonald's formulas

Established the Boson-Fermion correspondence and Weyl character formula for Macdonald polynomials

Abstract

This is a paper about $c$-functions and Macdonald polynomials. There are $c$-function formulas for $E$-expansions of $P_\lambda$ and $A_{\lambda+\rho}$, principal specializations of $P_\lambda$ and $E_\mu$, for Macdonald's constant term formulas, and for the norms of Macdonald polynomials. Most of these follow from the creation formulas for Macdonald polynomials, providing alternative proofs to several results from Macdonald (2003). In addition, we prove the Boson-Fermion correspondence in the Macdonald polynomial setting and the Weyl character formula for Macdonald polynomials.