In general, a function $f(\cdot)$ defined at points $x_1,x_2,\dots, x_k$, the $(k − 1)$th-order divided difference is defined by the recurrence relation:
$$ f[x_1,x_2,\dots...x_k]=\frac{f[x_2,\dots...x_{k-1},x_k]-f[x_1,x_2,\dots...x_{k-1}]}{x_k-x_1}, $$
with $f[x]=f(x)$. Is there a way to express analytically the $(k − 1)$th-order divided difference in terms only of divided differences of order 1?
Thanks in advance!