Two families of constant term identities

Abstract

In 1985, Bressoud and Goulden derived the formula for the constant term in $$\prod _{(i,j)\in T} \frac{x_j}{x_i} \prod _{0\le i<j \le n}(\frac{x_i}{x_j})_{a_i}(\frac{qx_j}{x_i})_{a_j-1}$$ , 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.