satisfying the recurrence [@rm3; @rm5; @t1; @rm7; @nex2]: $$P_{-1}(\omega )=0, \quad  P_0(\omega )=1,$$

$$\label{polrec}P_{n+1}(\omega )=(\omega -a_n)P_n(\omega )-b_n^2P_{n-1}(\omega
),$$

and $$\label{pola}a_n=\frac{\int_{-\infty }^{+\infty }\omega P_n^2(\omega )A_p(
{\bf k},\omega )d\omega }{\int_{-\infty }^{+\infty }P_n^2(\omega )A_p({\bf k}
,\omega )d\omega },$$ $$\label{polb}b_{n+1}^2=\frac{\int_{-\infty }^{+\infty }P_{n+1}^2(\omega )A_p(
{\bf k},\omega )d\omega }{\int_{-\infty }^{+\infty }P_n^2(\omega )A_p({\bf k}
,\omega )d\omega }.$$

Here we have used the nonnormalized form of the polynomials $\int_{-\infty }^{+\infty }P_n(\omega )P_s(\omega )A_p({\bf 
k},\omega )d\omega =\delta _{ns}(\prod_{m=1}^{m=n}b_m)^2$.

Comparing Eqs.(\[cf1\]) and (\[gineq\]) we see that the self energy $
\Sigma ({\bf k},z)$ is the continued fraction similar to $G({\bf 
k},z)$ . Thus we can introduce the spectral density

$$\rho ({\bf k},\omega )=-{\rm Im}[\Sigma ({\bf k},\omega +\imath 0^{+})]/\pi$$ and the set of polynomials $\Pi _n(\omega )$ with the recurrence analogous to (\[polrec\]): $$\Pi _n(\omega )=(\omega -a_n)\Pi _{n-1}(\omega )-b_n^2\Pi _{n-2}(\omega
),\quad \quad   \Pi _0(\omega )=1, \quad  \Pi _{-1}(\omega )=0,$$ where $$b_1^2=\int_{-\infty }^\infty \rho ({\bf k},\omega )d\omega ,\quad a_1=\frac 1
{b_1^2}\int_{-\infty }^{+\infty }\omega \rho ({\bf k},\omega )d\omega ,$$ $$\label{polab}a_{n+1}=\frac{\int_{-\infty }^{+\infty }\omega \Pi _n^2(\omega
)\rho ({\bf k},\omega )d\omega }{\int_{-\infty }^{+\infty }\Pi _n^2(\omega
)\rho ({\bf k},\omega )d\omega },\quad b_{n+1}^2=\frac{\int_{-\infty
}^{+\infty }\Pi _n^2(\omega )\rho ({\bf k},\omega )d\omega }{\int_{-\infty
}^{+\infty }\Pi _{n-1}^2(\omega )\rho ({\bf k},\omega )d\omega },\quad n\geq
1.$$

On the other hand, we have from 