$$f^{(m)}(z)=H_n(\{\mu_k\lambda_k^m\},\{\lambda_k\},h^{(m)};z)
+O(z^{2n-m}),\quad m=\overline{0, 2n-1}, \quad z\to 0,
 \label{2n-bis-interpol}$$ the following conditions on the so-called *generalized power sums* (*moments*) $S_m$ should be satisfied: $$\label{SRS} S_m:=\sum_{k=1}^n \mu_k \lambda_k^m=s_m,\qquad
m=\overline{0,2n-1}.$$

The system (\[SRS\]) with unknown $\mu_k$, $\lambda_k$ and given $s_m$ is well known as *the discrete moment problem*. Classical works of Prony, Sylvester, Ramanujan and papers of many contemporary researchers are devoted to the problem of its solvability (see [@Prony; @Sylvester; @Ramanujan; @Lyubich; @Lyubich2; @Kung1]). Note that the system (\[SRS\]) is bound up with Hankel forms, orthogonal polynomials, continued fractions, Gaussian quadratures and Padé approximants (a detailed review of these connections is given in [@Lyubich; @Lyubich2] and also in Section \[par2\]).

Suppose that the system (\[SRS\]) is solvable. Then, following [@Lyubich], we call the system and its solution *regular* if all $\lambda_k$ are pairwise distinct and all $\mu_k$ are not vanishing. In the case of regular systems (\[SRS\]) we call the problem of $2n$-multiple interpolation (\[2n-interpolation\]) *regularly solvable*. One of the methods for solving regular systems (\[SRS\]) is due to Prony [@Prony]. Consider the following product of determinants: $$\label{equ1} \left|
\begin{array}{ccccc}
1 & 0 & 0 & \ldots & 0\\
0 & \mu_1 & \mu_2 & \ldots & \mu_n\\
0 & \mu_1\lambda_1 & 