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We derive a recursion relation of the Feynman diagrams of the effectiveaction for the third Legendre transformation in case of the bosonic fieldtheory with cubic interaction. We apply the recursion relation to obtain theFeynman diagrams of the effective action for the third Legendre transformationup to the five-loop order. The three-particle irreducibility of the Feynmandiagrams of the effective action for the third Legendre transformation is shownby induction.

Recursion Relation for the Feynman Diagrams of the Effective Action for the Third Legendre Transformation