Let $n \in \mathbb{N}$, let $A \subset \mathbb{R}$, and let $x_0,x_1,\dots,x_n \in A$ be pairwise distinct. Let $f: A \to \mathbb{R}$ be a function. Define the Lagrange basis polynomials $\ell_i \in \mathbb{R}[t]$, for $0 \le i \le n$, by
be the Lagrange interpolation polynomial. Let $V_f(t) := \sum_{k=0}^{n} a_k t^k$ be the polynomial whose coefficient vector $(a_0,\dots,a_n) \in \mathbb{R}^{n+1}$ solves the Vandermonde system
for every $i \in \{0,1,\dots,n\}$. For integers $i,j$ with $0 \le i \le j \le n$, define the Newton divided differences $f[x_i,\dots,x_j] \in \mathbb{R}$ recursively by