I am trying to derive finite difference schemes from difference calculus. This is used in partial differential equations. For example if we take the first order one way wave equation
$$ u_t + u_x = 0 $$
We can create a discrete form to approximate the solution, where v is our discrete form. h is our spatial step and k is our time step
$$ \frac{v^{n+1}_m - v^{n}_m}{k} + \frac{v^{n}_{m+1} - v^{n}_{m-1}}{h} = 0 $$
This a first order accurate scheme in space and time. Difference notation can be used to simplify the expression as well as create higher order approximations. The above terms can be rewritten as
$$ \delta_0 v_n + \delta_0 v_m = 0 $$
The first order center difference $$\delta_0 v_m = \frac{v_{m+1} - v_{m-1}}{2h}$$
Second order difference $$\delta^2 v_m = \frac{v_{m+1} - 2V_m + v_{m-1}}{h^2}$$
In the book there is an example that shows
$$(1 - \frac{h^2}{6} \delta^2) \delta_0 v_m = f_m$$
This term can be written in a discrete form as
$$ \frac{-v_{m+2} + 8v_{m+1} -8v_{m-1} + v_{m-2}}{12h} = f_m$$
I am not sure how the second term is derived from the first term. Specifically $\delta^2 \delta_0 v_m$. Some notation was also given from the book but I haven't been able to find more information online for difference notation and difference calculus.