Sometimes it is useful to swap the order of the summations in a double summation. This isn’t completely obvious when the limits of one sum are dependent on another. Here’s the formula:
Note that inside the sum, we always have . On the left, i runs from h to k, and j runs from i to k (i.e. and ). On the right, i runs from h to j, and j runs from h to k (i.e. and ). Those are all summarised by .
A particularly common case is h=0 and k=n:
Notice here that within the sum .
Another common one is h=1 and k=n:
Notice in this case that within the sum .
To really illustrate this, let’s take the following simple example: h=1 and k=3 with . It doesn’t matter what φ is; I’ve only chosen this for compactness of example. First, let’s expand the left:
Now the right:
All that’s changed is the order.
I used the term ‘triangular sum’ in the title because of this: