Inverse of Vandermonde Matrix/Eisinberg Formula

Theorem
Let:

Assume $W_n$ has a matrix inverse $W_n^{-1} = \begin{bmatrix} d_{ij} \end{bmatrix}$.

Assume $a_n=\map {e_0} {x_1,\ldots,x_n} = 1$.

Then:

Lemma 1
Given values $z_1,\ldots,z_{p+1}$ and $1 \leq m \leq p$, then:

Lemma 2
Let $X = \set { x_1,\ldots,x_n }$ and $ {\mathbf u} = x_j$ for some $j=1,\ldots,n$.

Then:

Proof of Lemma 2:

Let $S$ denote the left side of (3).

Let $U = X \setminus \set { {\mathbf u} }$.

Then:

Proof of the Theorem

Historical Note
The Knuth Vandermonde inverse formula requires $n^2$ symmetric functions $\map {e_m} { \set {x_1,\ldots,x_n} \setminus \set {x_j} }$. Eisinberg and Picardi (1981) Vandermonde inverse formula (1) above is perhaps the first to use just $n$ elementary symmetric functions. The formula was revisited in Eisinberg and Fedele (2005), providing a concise proof without IBM Selectric typewriter fonts. Key identity (3) in Lemma 2 above is used in both references, isolated in Eisinberg, Franz and Pugliese (1998) as identity (8).