Generalized weight functions and the Macdonald polynomials

TL;DR

This paper introduces a generalized weight function related to Macdonald polynomials, presents conjectures and partial proofs for related identities, and explores orthogonal polynomials in the context of quantum many-body systems.

Contribution

It proposes a new weight function generalizing the ground state wave function and conjectures the orthogonality and properties of Macdonald polynomials with respect to this weight.

Findings

Conjectured constant term identities involving the new weight function.

Defined orthogonal polynomials conjectured to be Macdonald polynomials.

Partial proofs provided for special cases of the conjectures.

Abstract

A weight function which $q$-generalizes the ground state wave function of the multi-component Calogero-Sutherland quantum many body system is introduced. Conjectures, and some proofs in special cases, are given for a constant term identity involving this function. A Gram-Schmidt procedure with respect to the inner product associated with the weight function is used to define orthogonal polynomials in one of the components, which are conjectured to be the Macdonald polynomials $P_\kappa(w_1,\dots,w_{N_0};qt^p,t)$, and a proof is given in a special case. Conjectures are also given for an adjoint property of the Macdonald operator with respect to the inner product associated with the weight function, and the normalization of the Macdonald polynomial with respect to the same inner product.