  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 $SL(n,\mathbb{C})$.
