I have rephrased my original question to be clearer:
From my understanding, a multipole expansion is used to approximate the potential from a cluster of points, and can be evaluated at any point sufficiently far away. Suppose you have a cluster of points in box A which is centered at some point in space Q. The potential from these points can be approximated by the multipole potential evaluated at some point P using:
Where the primed variables are the coordinates P - Q.
Now suppose we want to find the multipole potential for the parent of box A. Of course, we could start over by finding the cluster of points in the parent box and recomputing the multipole expansion. But surely, we want to reduce computation time so it is better to form the parent multipole expansion from the parent’s children.
If we find the multipole expansion given earlier for all the children, the potential at some point P due to the parent will of course be the sum of the children’s multipole expansions evaluated at P. (note: the math is correct for this, and it makes complete sense because potentials follow the superposition principle)
However, the translation theorem commonly used to convert children multipole expansions to their parent is given by:
It seems obvious one could just add the multipole expansions to form the parent expansion. Why is a different translation theorem used when converting children expansions to their parent expansion? Is this translation theorem applied in a different setting?