Prove/disprove the following equality (sum of inverse difference products)

Question

$$\sum_{x\in S}\frac{1}{\prod_{y\in S, y\neq x}(x-y)} = 0$$

Here, S is a finite subset of real numbers.

Answer 1

Let $n=|S|\geq 2$. Given an arbitrary set of $n$ real values $t_1,\ldots,t_n$, there is a unique$^{(*)}$ polynomial with degree $\leq(n-1)$ such that $p(s_1)=t_1,\ldots,p(s_n)=t_n$ and this polynomial can be constructed via Lagrange interpolation:

$$ p(x) = \sum_{k=1}^{n} t_k \prod_{j\neq k}\frac{x-s_j}{s_k-s_j}. $$ We may notice that $$ \sum_{k=1}^{n} t_k \prod_{j\neq k}\frac{1}{s_k-s_j}$$ is the coefficient of $x^{n-1}$ in $p(x)$.
If $t_1=\ldots=t_n=1$, the interpolating polynomial is $p(x)=1$ and $\left[x^{n-1}\right]p(x)=0$.

^{(*) Incidentally, this also proves that the Vandermonde matrix associated to $s_1,\ldots,s_n$ is invertible.}