Now there is something about the Legendre differential equation that drove me crazy. What is up with the l(l+1) !!!

To understand why let’s take this form of the LDE and arrive at the above:

If we do a power series expansion and following the same steps as the previous post, we end up with the following recursion relation.

or

Here’s the deal: We want a convergent solution for our differential solution. This means that as .

Hence we obtain that