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We establish an algorithm which computes formulae for the CR GJMS operators,the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors.When applied to torsion-free pseudo-Einstein contact forms, this algorithm bothgives an explicit factorisation of the CR GJMS operators and the$P^\prime$-operator, and shows that the $Q^\prime$-curvature is constant, withthe constant explicitly given in terms of the Webster scalar curvature. We alsouse our algorithm to derive local formulae for the $P^\prime$-operator and$Q^\prime$-curvature of a five-dimensional pseudo-Einstein manifold. Comparisonwith Marugame's formulation of the Burns--Epstein invariant as the integral ofa pseudohermitian invariant yields new insights into the class of localpseudohermitian invariants for which the total integral is independent of thechoice of pseudo-Einstein contact form.

The $P^\prime$-operator, the $Q^\prime$-curvature, and the CR tractor calculus