$h$ and thus one has a little chance to obtain more or less general interpolation formulas of the overdetermined type.

For example, if a solvable system of the form (\[SRS\]) is supposed to be consistent with the next equation $S_{2n}=s_{2n}$, then the coefficient $f_{2n}$ cannot be chosen arbitrarily as it depends on the parameters $s_0,\ldots s_{2n-1},h_{2n}$ in a certain way. Indeed, the following generalized Newton’s formula, connecting the coefficients $g_m$ of the generating polynomial $G_n$ and the moments $S_{v+m}$, is well-known: $$\label{corollary_form_Newton++++} \sum_{m=0}^{n} S_{v+m}{g}_{m}=0,
\qquad v=0,1,\ldots.$$ Therefore the values $s_m=s_m(h,f)$ (see (\[ssmm\])) of the moments $S_m$ with ${m>2n-1}$ (simultaneously with the coefficients $f_{m}=s_m h_m$) in a solvable overdetermined system of the form (\[SRS\_M\]) are uniquely determined from the system (\[SRS\]) (we suppose that $h$ is fixed). A similar situation arises when one obtains formulas for numerical differentiation and extrapolation (see the corresponding sections below). In these problems the sequences $\{s_m\}$ have a certain arithmetic structure and do not satisfy (\[corollary\_form\_Newton++++\]) for $v=n$ and $S_{2n}=s_{2n}$, i.e. one gets solvable systems of the form (\[SRS\]) with $M=2n$ although adding one more equation of the required form leads to an inconsistent system (see, for example, (\[r\_n\_diff\])).

As regards the underdetermined moment systems (when 