Symmetric function generalizations of the $q$-Baker--Forrester ex-conjecture and Selberg-type integrals

TL;DR

This paper introduces two symmetric function generalizations of the $q$-Baker--Forrester ex-conjecture, extending known identities involving Macdonald polynomials and complete symmetric functions.

Contribution

It presents novel symmetric function generalizations of the $q$-Baker--Forrester ex-conjecture, expanding the scope of Selberg-type integrals and constant term identities.

Findings

Derived a $q$-Baker--Forrester type constant term identity involving Macdonald polynomials.

Established a complete symmetric function generalization of KNPV's result.

Extended the framework of Selberg-type integrals with symmetric function generalizations.

Abstract

It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the $q$-Morris constant term identity. This conjecture was proved and extended by K\'{a}rolyi, Nagy, Petrov and Volkov in 2015. In this paper, we obtain two symmetric function generalizations of the $q$-Baker--Forrester ex-conjecture. These includes: (i) a $q$-Baker--Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result.