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

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 , . . . , v n ) in the case when only one v i = 0. This conjecture was first proved by Károlyi, 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.

A symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem

“…Recently, we got a recursion for D v,v + (a) if v is a weak composition with unique largest part [12]. Later, we obtained a recursion for D v,v + (a) for arbitrary nonzero weak composition v in [13]. If v is a weak composition with equal nonzero parts, for example v = (2, 2, 0 n−2 ), it appears that the expression for D v,v + (a) is no longer a product.…”