TITLE: Is there a way to interpret summation by parts as integration by parts with counting measure?
QUESTION [5 upvotes]: I find it difficult to remember the different forms of summation by parts: where the indices begin, end, whether to take forward/backward differences, etc. For example, Wikipedia has one form
$$\sum_{k=m}^n f_k(g_{k+1}-g_k) = \left[f_{n+1}g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}(f_{k+1}- f_k)$$
and I've seen another
$$\sum_{k=m}^n f_k(g_k-g_{k-1}) = [f_n g_n - f_mg_{m-1}] - \sum_{k=m}^{n-1} g_k (f_{k+1} - f_k)$$
Even though I can derive one from the other, it takes some time. So I'm wondering if all this can be easily seen in a general setting with counting measure. Or perhaps if there are some nice heuristics which can help me remember the different forms quickly.

REPLY [2 votes]: The same proof idea should work. For example, starting with
$$\Delta(f_n g_n) = f_{n+1} g_{n+1} - f_n g_n $$
and using any of the usual tricks to convert this into, e.g.,
$$\ldots = f_{n+1} \Delta g_n + g_n \Delta f_n $$
and then you can just add up both sides.
With a little practice, you can quickly go from some $f_n \Delta g_{n+7}$ and then figure out what you need to add in to "complete the delta"; e.g. what you need to add to make it $\Delta(f_n g_{n+7})$. (Why is a good delta to complete it to? Because both have $f_n g_{n+7}$ with a negative coefficient)
I'm used to $\Delta$ always meaning a forward difference; if you pick one meaning for $\Delta$ and stick with it, it's easier to work with these things.