Write $X_{B,b} = \{\alpha \in \mathbb{Z}^B : \sum_j \alpha_j = b\}$. Now using the convention $0^0 \equiv 1$, define the matrix $W_{\alpha, \alpha'} := \alpha^{\alpha'}$. For arbitrary $f:X_{B,b} \rightarrow \mathbb{K}$ we can write $f_\alpha \equiv \sum_{\alpha'} c_{\alpha'}$ from which it follows that $c = W^{-1}f$. The facts that this procedure is well-defined, and that $W$ possesses an inverse, follow from a result in multivariate interpolation assuring us that the Lagrange interpolation problem on $X_{B,b}$ is "poised".
Actually, although generic discrete point sets admit a specific multivariate Lagrange interpolation protocol that satisfies many desirable properties, only $X_{B,b}$ does it so beautifully. As a result, we obtain a Lagrange interpolation: $f_{\mathfrak{I}}(x) := \sum_\alpha (W^{-1}f)_\alpha x^\alpha$ which satisfies $f_{\mathfrak{I}}(\alpha) = f(\alpha)$.
You can use this to define differencing schemes on the hexagonal grid by considering $B = 3$. An example of the interpolation is shown.
