In Newton's divided differences interpolation method, the $k$ th divided difference relative to the points $\left( x_{i},f(x_{i}) \right), \left( x_{i+1},f(x_{i+1}) \right), \left( x_{i+2},f(x_{i+2}) \right), \ldots, \left( x_{i+k},f(x_{i+k}) \right)$ is deffined as
$$f_i^{[i+k]}=f\left[x_{i}, x_{i+1}, \ldots, x_{i+k-1}, x_{i+k}\right]=\frac{f\left[x_{i+1}, x_{i+2}, \ldots, x_{i+k}\right]-f\left[x_{i}, x_{i+1}, \ldots, x_{i+k-1}\right]}{x_{i+k}-x_{i}}$$
with $f_i^{[i]}=f[x_i]=f(x_i)$ and $i,k\in \{0,1,2,3,...\} $.
Are this divided differences symmetrical? That is, do they verify the following?
$$f_i^{[i+k]}\overset{?}{=}f_{i+k}^{[i]}$$