\mu_2\lambda_2 & \ldots & \mu_n\lambda_n\\
\ldots & \ldots & \ldots & \ldots & \ldots\\
0 & \mu_1\lambda_1^{n-1} & \mu_2\lambda_2^{n-1} & \ldots & \mu_n\lambda_n^{n-1}\\
\end{array}
\right| \cdot \left|
\begin{array}{ccccc}
1 & \lambda & \lambda^2 & \ldots & \lambda^n\\
1 & \lambda_1 & \lambda_1^2 & \ldots & \lambda_1^n\\
\ldots & \ldots & \ldots & \ldots & \ldots\\
1 & \lambda_n & \lambda_n^2 & \ldots & \lambda_n^n\\
\end{array}
\right|.$$ By regularity, the former of them does not vanish and the latter does only for $\lambda=\lambda_k$ (as a Vandermonde determinant). On the other hand, direct multiplication of the determinants and taking into account (\[SRS\]) give the following determinant, which is a polynomial of $\lambda$: $$\label{G_n} G_n(\lambda):=\sum_{m=0}^n g_{m} \lambda^{m}= \left|
\begin{array}{ccccc}
1 & \lambda & \lambda^2 & \ldots & \lambda^n\\
s_0 & s_1 & s_2 & \ldots & s_n\\
s_1 & s_2 & s_3 & \ldots & s_{n+1}\\
\ldots & \ldots & \ldots & \ldots & \ldots\\
s_{n-1} & s_n & s_{n+1} & \ldots & s_{2n-1}\\
\end{array}
\right|.$$ We call $G_n$ [*a generating polynomial*]{} (for functional properties of such polynomials, including orthogonality, completeness, etc., see [@Lyubich]). Consequently, the numbers $\lambda_k$ are the simple roots of the generating polynomial $G_n$. If the equation $G_n(\lambda)=0$ is solved, 