A unified elementary approach to the Dyson, Morris, Aomoto, and Forrester constant term identities

Abstract

We introduce an elementary method to give unified proofs of the Dyson, Morris, and Aomoto identities for constant terms of Laurent polynomials. These identities can be expressed as equalities of polynomials and thus can be proved by verifying them for sufficiently many values, usually at negative integers where they vanish. Our method also proves some special cases of the Forrester conjecture.

A new approach to constant term identities and Selberg-type integrals

“…It involves the same combinatorics applied when k ≥ a + 1, in which case A 0 is an ordinary set. The extension of the result that includes all non-negative integers k depends on the following rationality lemma, inspired by [20,Proposition 2.4]. Expanding the degree zero part of n j=1 (qx j ) a+χ( j≤m) (1/x j ) b n 0 <i< j≤n (1 − q k x i /x j )(1 − q k+1 x j /x i ) into a sum of monomial terms and applying the above lemma to each such term individually, we find that there is a rational function R ∈ Q(q)(z) depending only on the parameters n, m, n 0 , a, b such that…”

“…see also [20]. Here the full capacity of Theorem 2.4 can be exploited with a minimum amount of computation.…”

On restricted sumsets over a field

“…In order to prove Proposition 2.2, we also need the following result (see Lemma 2.1 and Corollary A.1 in [8]). Lemma 2.3 (Gessel-Lv-Xin-Zhou).…”

A Laurent Series Proof of the Habsieger-Kadell $q$-Morris Identity

“…Dyson's ex-conjecture has been proved by many authors using different methods. See, e.g., [8,10,11,21,22]. Many variations of Dyson's ex-conjecture have been found, such as the famous Macdonald constant term conjectures [6,17].…”

“…Our approach is by extending the proof of the Aomoto identity in [8]. The basic idea is to regard both sides of (1.3) as polynomials in q a of degree at most d = nb+m+l.…”