The divided difference of a function \(f\) with respect to centers
\(z_1, \ldots, z_{k+1}\) is defined recursively as:
$$
f[z_1,\ldots,z_{k+1}] = \displaystyle
\frac{f[z_2,\ldots,z_{k+1}] - f[z_1,\ldots,z_k]}{z_{k+1}-z_1},
$$
with base case \(f[z_1] = f(z_1)\) (that is, divided differencing with
respect to a single point reduces to function evaluation).
A notable special case is when the centers are evenly-spaced, say, \(z_i =
z+ih\), \(i=0,\ldots,k\) for some spacing \(h>0\), in which case the
divided difference becomes a (scaled) forward difference, or equivalently a
(scaled) backward difference,
$$
k! \cdot f[z,\ldots,z+kh] = \displaystyle
\frac{1}{h^k} (F^k_h f)(z) =
\frac{1}{h^k} (B^k_h f)(z+kh),
$$
where we use \(F^k_h\) and \(B^k_v\) to denote the forward and
backward difference operators, respectively, of order \(k\) and with
spacing \(h\).