A family of q-Dyson style constant term identities

By generalizing Gessel–Xinʹs 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 Sillsʹ as special cases and generalizing Stembridgeʹs first layer formulas for characters of SL ( n , C ) .