Two Families of Constant Term Identities

Abstract

In 1985, Bressoud and Goulden derived the formula for the constant term in $$\begin{gathered} {\prod }_{(i,j)\in T}\frac{x_j}{x_i} \\ {\prod }_{0\le i<j\le n}(\frac{x_i}{x_j}{)}_{a_i}(\frac{q{x}_j}{x_i}{)}_{a_j-1} \end{gathered}$$, where $T\subseteq \{(i,j)\mid 0\le i<j\le n\}$. This result implies the Andrews' $q$-Dyson identity. In 2006, Gessel and Xin proved the $q$-Dyson identity by considering both sides of the equality as polynomials in $q^{a_0}$. We use this approach to determine the coefficients of $x_0/x_1$ and $x_0/x_2$ in Laurent polynomials studied by Bressoud and Goulden.

arXiv commentarXiv admin note: text overlap with arXiv:math/0412339 by other authors