Consider a Taylor expansion of a function $f$ of $N$ variables $\mathbf{x}$, about $\mathbf{x}=\mathbf{0}$:

$$ f(\mathbf{x}) = f(\mathbf{0})+\sum_i^Nc_ix_i + \frac{1}{2!} \sum_{i,j}^N c_{ij}x_ix_j + \frac{1}{3!} \sum_{i,j,k}^N c_{ijk}x_ix_jx_k + \frac{1}{4!} \sum_{i,j,k,l}^N c_{ijkl}x_ix_jx_kx_l + \cdots$$

where indices $ijkl$ represent different variables, $c_{ij}$ are unknown constants representing 2nd derivatives with respect to $x_i$ and $x_j$, $c_{ijk}$ are unknown constants representing 3rd derivatives, etc.

Where can I find finite difference equations for the constants $c$ (i.e., the derivatives), at least up to 4th order?

Furthermore, how can I derive finite difference equations for these constants?


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