A recursion for a symmetric function generalization of the $q$-Dyson constant term identity

TL;DR

This paper generalizes a recursive formula for a symmetric function-based constant term identity related to the $q$-Dyson theorem, extending previous results to all compositions.

Contribution

It introduces a recursion for the constant term identity applicable to all compositions, broadening the scope of earlier specific cases.

Findings

Established a recursion for the constant term for all compositions

Extended previous results from special cases to general compositions

Provided a unified approach to the symmetric function generalization

Abstract

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the $q$-Dyson constant term identity or the Zeilberger--Bressoud $q$-Dyson theorem. The non-zero part of Kadell's orthogonality conjecture is a constant term identity indexed by a weak composition $v=(v_1,\dots,v_n)$ in the case when only one $v_i\neq 0$. This conjecture was first proved by K\'{a}rolyi, Lascoux and Warnaar in 2015. They further formulated a closed-form expression for the above mentioned constant term in the case when all the parts of $v$ are distinct. Recently we obtain a recursion for this constant term provided that the largest part of $v$ occurs with multiplicity one in $v$. In this paper, we generalize our previous result to all compositions $v$.