G a 1 , − $_{1}$( x | x $^{′}$) = F $^{a}$( x | x $^{′}$) , G a − 1 , $_{1}$( x | x $^{′}$) = F $^{a}$( x | x $^{′}$) , G a − 1 , − $_{1}$( x x $^{′}$) = − G $^{a}$( x $^{′}$| x ) . (10) Λ$_{αα}$ ′ ( x | x $^{′}$) = G f (0) αα ′ ( x | x $^{′}$) + Z$_{αα}$ ′ ( x | x $^{′}$) , (13) G (0) ir n [ α$_{1}$, x$_{1}$ ; ... ; α$_{2}$$_{n}$, x$_{2}$$_{n}$ ] = 〈 T f α$_{1}$ x$_{1}$ ...f α$_{2}$ n x$_{2}$ n 〉 ir 0 = (11) = δ$_{x}$$_{1}$$_{x}$$_{2}$...δ$_{x}$$_{1}$$_{x}$$_{2}$$_{n}$ 〈 T f α$_{1}$ σ$_{1}$ ( τ$_{1}$ ) ...f α$_{2}$ n σ$_{2}$ n ( τ$_{2}$$_{n}$ 〉 ir 0 , 〈 T f α$_{1}$ σ$_{1}$ ( τ$_{1}$ ) f α$_{2}$ σ$_{2}$ ( τ$_{2}$ ) f α$_{3}$ σ$_{3}$ ( τ$_{3}$ ) f α$_{4}$ σ$_{4}$ ( τ$_{4}$ ) 〉 ir 0 = (12) = T f α$_{1}$ σ$_{1}$ ( τ$_{1}$ ) f α$_{2}$ σ$_{2}$ ( τ$_{2}$ ) f α$_{3}$ σ$_{3}$ ( τ$_{3}$ ) f α$_{4}$ σ$_{4}$ ( τ$_{4}$ ) 0 − − [ 〈 T f α$_{1}$ σ$_{1}$ ( τ$_{1}$ ) f α$_{2}$ σ$_{2}$ ( τ$_{2}$ ) 〉 0 〈 T f α$_{3}$ σ$_{3}$ ( τ$_{3}$ ) f α$_{4}$ σ$_{4}$ ( τ$_{4}$ ) 〉 0 + + T f α$_{1}$ σ$_{1}$ ( τ$_{1}$ ) f α$_{4}$ σ$_{4}$ ( τ$_{4}$ ) 0 T f α$_{2}$ σ$_{2}$ ( τ$_{2}$ ) f α$_{3}$ σ$_{3}$ ( τ$_{3}$ ) 0 − − 〈 T f α$_{1}$ σ$_{1}$ ( τ$_{1}$ ) f α$_{3}$ σ$_{3}$ ( τ$_{3}$ ) 〉 0 〈 T f α$_{2}$ σ$_{2}$ ( τ$_{2}$ ) f α$_{4}$ σ$_{4}$ ( τ$_{4}$ ) 〉 $_{0}$] . G c αα ′ ( x | x $^{′}$) = G c (0) αα ′ ( x | x $^{′}$) + (15) + ∑ α$_{1}$α$_{2}$ ∑ 12 G c (0) $_{αα$_{1}$}$( x | 1)Σ$_{α}$$_{1}$$_{α}$$_{2}$(1 | 2) G c α$_{2}$α ′ (2 | x $^{′}$) . Σ$_{αα}$ ′ ( x | x $^{′}$) = V $^{2}$αα $^{′}$Λ$_{αα}$ ′ ( x | x $^{′}$) (14) G f αα ′ ( x | x $^{′}$) = Λ$_{αα}$ ′ ( x | x $^{′}$) + (16) + V 2 ∑ α$_{1}$α$_{2}$ ∑ 12 α$_{1}$α$_{2}$ Λ$_{αα}$$_{1}$( x | 1) G c (0) α$_{1}$$_{α$_{2}$}$(1 | 2) G f α$_{2}$α ′ (2 | x $^{′}$) . V 2 ∑ α$_{1}$α$_{2}$ ∑ 12 α$_{1}$α$_{2}$G c (0) $_{αα$_{1}$}$( x | 1) G f (0) α$_{1}$$_{α$_{2}$}$(1 | 2) G c (0) α$_{2}$α ′ (2 | x $^{′}$) , V 4 2 ∑ α$_{1}$...α$_{4}$ ∑ 1 ... 4 α$_{1}$α$_{2}$α$_{3}$α$_{4}$G c (0) $_{αα$_{1}$}$( x | 1) G c (0) α$_{2}$$_{α$_{3}$}$(2 | 3) × × G c (0) α$_{4}$α ′ (4 | x $^{′}$) 〈 T f α$_{1}$ 1 f − α$_{2}$ 2 f α$_{3}$ 3 f − α$_{4}$ 4 〉 ir 0 1 2 ∑ α$_{1}$α$_{2}$ ∑ 12 α$_{1}$α$_{2}$G c α$_{1}$$_{α$_{2}$}$(1 | 2) 〈 T f α $_{x}$f − α$_{1}$ 1 f α$_{2}$ 2 f − α ′ x ′ 〉 ir 0 , − 1 8 ∑ α$_{1}$...α$_{4}$ ∑ 1 ... 4 α$_{1}$α$_{2}$α$_{3}$α$_{4}$G c α$_{1}$$_{α$_{2}$}$(1 | 2) G c α$_{3}$$_{α$_{4}$}$(3 | 4) × × 〈 T f α $_{x}$f − α$_{1}$ 1 f α$_{2}$ 2 f − α$_{3}$ 3 f α$_{4}$ 4 f − α ′ x ′ 〉 ir 0 , correspondingly. correspondingly. Between the diagrams for the one-particle f − propaga- tor there are strong and weak connected ones. The weak connected diagrams can be separated in two parts by cut- ting one propagator line. The sum of all strong connected The contribution of the diagrams Fig. 3 b) and c) is the following: On the Fig. 3 the skeleton diagrams for the correla- tion function Λ$_{αα}$ ′ ( x | x $^{′}$) are depicted . They demonstrate impossibility to formulate Dyson-type equation for this function. The number of skeleton diagrams depicted on the Fig. 3 for the correlation function is infinite. The contributions of perturbation theory for f − elec- tron propagator are depicted on the Fig. 2 The contri- butions of diagrams Fig. 2 b) and c) are equal to correspondingly. It demonstrates the dependence of the diagrams from the charge quantum number α . V 2 ∑ α$_{1}$α$_{2}$ ∑ 12 α$_{1}$α$_{2}$G f (0) $_{αα$_{1}$}$( x | 1) G c (0) α$_{1}$$_{α$_{2}$}$(1 | 2) G f (0) α$_{2}$α ′ (2 | x $^{′}$) , V 2 2 ∑ α$_{1}$α$_{2}$ ∑ 12 α$_{1}$α$_{2}$G c (0) α$_{1}$$_{α$_{2}$}$(1 | 2) 〈 T f α $_{x}$f − α$_{1}$ 1 f α$_{2}$ 2 f − α ′ x ′ 〉 ir 0 , Unfortunate the Dyson-type equations far correlation function Λ$_{αα}$ ′ and mass operator Σ$_{αα}$ ′ don’t exist. There- fore the calculation of the c − and f − renormalized propa- gators needs the approximations based on the summation of special classes of diagrams. In equation (15) and (16) as in the previous equations which contain repeated indices (1 , 2) is supposed sum- mation by sites indices ( 1 , 2 ), spin indices ( σ$_{1}$, σ$_{2}$ ) and integration by time variables ( τ$_{1}$, τ$_{2}$ ) in the interval (0 , β ). As a result of applying these theorem we obtain for the renormalized conduction electron propagator the contri- butions depicted on the Fig. 1 The contributions of the diagrams Fig. 1 b) and e) are the following For n ≥ 3 the irreducible Green’s function contains in its right-hand part besides the products of one-particle propagators also their products with irreducible functions of smaller number of particles. There are present also the product of irreducible functions G (0) ir n$_{1}$ and G (0) ir n$_{2}$ with condition n$_{1}$ + n$_{2}$ = n and so on [10 − $^{13]}$. At the same time we can formulate the Dyson-type equa- tion for correlated electron propagator G $^{f}$: The analysis of the propagator diagrams permits us to formulate the following Dyson equation for uncorrelated electron propagator where in simplest two-particle case we have the following definition of the irreducible function A simple relation exists between these two functions: The strong connected part of the c − electron propa- gator without the external lines is determined by us as a mass operator for uncorrelated electrons. This quantity is denoted as Σ$_{αα}$ ′ ( x | x $^{′}$). where the function Z$_{αα}$ ′ ( x | x $^{′}$) contains the contribution of strongly connected diagram based on the irreducible many-particle Green’s functions. In the presence of strong correlations of f − electrons the (GWT) contains additional terms namely the irreducible one-site many-particle Green’s functions or Kubo cumu- lants of the form ( x = x , σ, τ ): diagrams for f − electron belong to the correlation func- tion which is denoted by us as Λ$_{αα}$ ′ ( x | x $^{′}$) function. The quantity Λ$_{αα}$ ′ ( x | x $^{′}$) is defined by the equation 3