Did anybody consider $\pi$-based finete differences, that is the operator $$\Delta_\pi f(x)=f(x+\pi)-f(x)$$ and its corresponding inverse operator? It seems for me that taking the step equal to $\pi$ simplifies many identities much.
For example,
$$\Delta_\pi \sin x = - 2 \sin x$$
$$\Delta_\pi \tan x = 0$$
$$\Delta_\pi^{-1} \sin x = -\frac 12 \sin x + C$$
$$\Delta_\pi^{-1} \cos x = -\frac 12 \cos x + C$$
$$\Delta_\pi^{-1} \tan x = \frac x\pi \tan x + C$$
I want a link to a paper if it is possible, or, at least the term used for such an operator.
