The procedure outlined in the previous section would be efficient if after calculating a finite number of coefficients $a_n,b_n,n\leq 
n_0$, we could appropriately approximate the part (infinite in our case) of the continued fraction $T_{n_0}$, which has not been calculated. In other words, we rewrite the expression (\[cf1\]) in the form $$\label{cf2}G({\bf 
k},z)=\frac{b_0^2}{z-a_0-}\frac{b_1^2}{z-a_1-}\cdots \frac{ 
b_{n_0}^2}{z-a_{n_0}-T_{n_0}({\bf k},z)},$$ and try to find a function $\tilde T_{n_0}$ (so called ”terminator”) that is close to $T_{n_0}$.

Various ways to construct such approximations are described in the literature on the recursion method (see Refs.[@rm5; @t1; @rm7] ). The asymptotic behavior of continued fraction coefficients is governed by the band structure and singularities of spectral density [@rm7]. The main asymptotic behavior depends on the band structure: $\{a_n\}$ and $
\{b_n\}$ converge towards limits in the single band case, oscillate endlessly in a predictable way in the multiband case. Damped oscillations are created by isolated singularities. The main point here is that an isolated simple pole produces [*exponentially*]{} damped contribution in $
\{a_n\},\{b_n\}$ , $n\rightarrow \infty $. For our case it means that the quasiparticle pole position and weight could be obtained with the high accuracy from finite number of coefficients, and the asymptotic behavior determines the 