Legendre transformations and equations for Green's functions

Higher Legendre transformations of the generating function of the Green's functions are used to obtain relations which have the meaning of many-particle equations. The second Legendre transformation gives two relations: the Edwards equation and the Bethe–Salpeter equation. The third Legendre transformation gives a system of three equations corresponding to the processes $3\to 1$, $3\to 2$, $3\to 3$.