In such a way we obtain the contribution of one line of skeleton diagram multiplied by the number of lines of skeleton diagrams. This number changes the coefficient before the skeleton diagram of Y ′ and new coefficients correspond to new contribution to Y ′ equal to the self- energy one. This quantity has the form of second term in right hand part of functional (19) for Y having the oppo- site sign. When we combine the both part of functional Y these quantities are reciprocally canceled. Finally we obtain to the first order for F the equation We use the Poisson equation for the ω$_{n}$ sums and write them as an integral where C is contour which surrounds in anti clock wise di- rection the poles of the function ( e $^{βz}$+1) − 1 in the points z = iω$_{n}$ = (2 n +1) π β i . The term in last equation propor- tional to T 2 is obtained by usual Sommerfeld technique. The details of such computation will be discussed in other place. tions it is impossible to formulate Dyson-type equations for Λ and Σ quantities. The results are appropriate as for normal and as the su- perconducting state. Unification of the investigation for the both phases was possible thanks the introducing of the notion of quantum charge number α and the rewrit- ten of the interaction Hamiltonian in such new form. From Fig. 3 it is clear that the simplest contribution that takes into account f − electron correlations is re- duced to first two terms of right-hand part of this figure. All the terms of Fig. 3 besides the last one and also other omitted diagrams like them are local with Fourier representation independent of momentum. These terms correspond to the structure of dynamical mean field the- ory. Last diagram of Fig. 3 and other more complicated diagrams with more number of irreducible Green’s func- tions depend of momentum and take in consideration of the space fluctuations. The local contributions take into account only of the fluctuations in time. We have demonstrated the transition of our diagram from superconducting to normal state by using the addi- tional conditions imposed on the charge quantum num- bers of which depend the dynamical quantities. The special investigation of vacuum diagram has been done after introducing the auxiliary interaction strength and integration by it of these diagram contributions. We have proved that this integrant is equal to the product of two matrices Σ$_{αα}$ ′ and G c αα ′ . Then we have introduced special functional in the form of skeleton diagrams and proved it coincidence with ther- modynamical potential. This expression has the property We have obtained the skeleton diagrams for Λ func- tion and demonstrated their dependence from irreducible many-particle Green’s functions G (0) ir n (1 , ..., 2 n ) with all values of n and also of c − electrons full propagators. Thanks the presence of these irreducible Green’s func- We introduced the notion of correlation function Λ$_{αα}$ ′ ( x | x $^{′}$) of f − electrons (see Fig. 3) which is the in- finite sum of strong connected irreducible Green’s func- tions and which contains the most important spin, charge and pairing fluctuations of the correlated f − electrons. This correlation function determines the mass operator Σ$_{αα}$ ′ ( x | x $^{′}$) (14) of the uncorrelated conduction electrons. The both these quantities Λ and Σ permit us to formu- late the Dyson equation for c − electrons (15) and Dyson- type equation (16) for f − electrons. These results are expressed in general form appropriate as for normal and as for superconducting state. We have developed the diagrammatic theory for PAM on the base of new conceptions proposed by us for strongly correlated electron systems. VI. CONCLUSIONS F = F$_{0}$ − 1 2 ∑ σ k ∫ C dze z 0 + e βz + 1 ln [ G c $^{(0)}$( z )Σ( k | z ) | T =0 − 1] , (39) F = F$_{0}$ − 1 2 β ∑ σ k ∑ ω n ln[ G c $^{(0)}$( iω$_{n}$ )Σ( k | iω$_{n}$ ) | T =0 − 1] e iω n 0 + = (38) = F$_{0}$ + 1 2 β ∑ σ k ∑ ω n ln[ iω$_{n}$ − ϵ$_{k}$ ] e iω n 0 + − 1 2 β ∑ σ k ∑ ω n ln[Σ( k | iω$_{n}$ ) | T =0 − ( G c $^{(0)}$( iω$_{n}$ )) − $^{1}$] e iω n 0 $^{+}$. 11