I have come across an equation that looks like
$$ f\!\left(t + \tau, x\right) = f\!\left(t, x - \xi\right) - f\!\left(t, x + \xi\right) $$
where $\tau$ and $\xi$ are both small. Every bone in my body says "convert that to a PDE", but when I do a series expansion, I find that not all my terms are linear in the small quantities:
$$ f\!\left(t, x\right) + \tau \partial_t f\!\left(t, x\right) + \mathcal{O}\!\left(\tau^2\right) = -2 \xi \partial_x f\!\left(t, x\right) + \mathcal{O}\!\left(\xi^2\right) $$
I understand how the series expansion works and I can see how this happened, but I don't know how to proceed. If everything were linear in small quantities, I'd know how to continue.
If I take a limit as $\tau$ and $\xi$ approach zero as-is, I'll just get $f\!\left(t, x\right) = 0$. That seems like a sad and unsatisfactory ending for such a nice equation. I'm hoping someone here has another idea.