I am trying to find an analytical way to describe the finite difference coefficients of various degrees of accuracy of centred difference schemes that approximate the second derivative. For example, a second order approximation is:
$$\frac{\partial^2}{\partial x^2} p(x) = \frac{p(x-h) - 2 p(x) + p(h+1)}{2h} + O(x^2)$$
so the first set of coefficients is $\{\frac{1}{2}, -1, \frac{1}{2} \}$. Likewise, you can extend this method to higher orders. I made a table of the first 8 sets of coefficients:
The only way I know to derive these is to use Lagrangian interpolation, and then solve a system of $n + 1$ equations (where $n$ is the order of accuracy). However, this doesn't really get you any closer to an analytical way of describing it. There might be another, more analytical approach (Taylor series expansions?), or a formula that computes this for arbitrary orders, but I don't know it.
I am fully aware of the possibility that such an analytical representation may not exist, but I'd be interested to know if there is a way to prove that.