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We describe an elementary algorithm for expressing, as explicit formulae intractor calculus, the conformally invariant GJMS operators due to C.R. Grahamet alia. These differential operators have leading part a power of theLaplacian. Conformal tractor calculus is the natural induced bundle calculusassociated to the conformal Cartan connection. Applications discussed includestandard formulae for these operators in terms of the Levi-Civita connectionand its curvature and a direct definition and formula for T. Branson'sso-called Q-curvature (which integrates to a global conformal invariant) aswell as generalisations of the operators and the Q-curvature. Among examples,the operators of order 4, 6, and 8 and the related Q-curvatures are treatedexplicitly. The algorithm exploits the ambient metric construction of Feffermanand Graham and includes a procedure for converting the ambient curvature andits covariant derivatives into tractor calculus expressions. This is partlybased on "Standard tractors and the conformal ambient metric construction" (A.Cap and A.R. Gover, math.DG/0207016), where the relationship of the normalstandard tractor bundle to the ambient construction is described.Comment: 42 pages. No figures. Record of changes: V1, 15 January 2002:  Original posting. V2, 17 January 2002: Changing comment fields. Leaving  abstract and text of article unchanged. V3, 1 February 2003: Minor changes  and typographical corrections throughout articl

Conformally Invariant Powers of the Laplacian, Q -Curvature, and Tractor Calculus