Explicit formulas for GJMS-operators and $Q$-curvatures

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincar'e-Einstein metrics and renormalized volume coefficients. As special cases, we find explicit formulas for conformally covariant third and fourth powers of the Laplacian. Moreover, we prove related formulas for all Branson’s $Q$-curvatures. The results settle and refine conjectural statements in earlier works. The proofs rest on the theory of residue families.