A Family of $q$-Dyson Style Constant Term Identities

By generalizing Gessel-Xin’s Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the $q$-Dyson product, including three conjectures of Sills’ as special cases and generalizing Stembridge’s first layer formulas for characters of $SL(n,\mathbb{C})$.