Inverse of Vandermonde Matrix/Eisinberg Formula

Theorem
Let:

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

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

Then:

Lemma 1
Given values $z_1, \ldots, z_{p + 1}$ and $1 \le m \le p$, 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 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).