Fourth derivative of a Bessel function

Question

I am given with a relation \begin{equation} \frac{d^2}{dx^2}(P_l(x))=\frac{1}{2}\sum_{n=(0,1),2}^{l-2}(l-n)(l+n+1)(2n+1)P_n(x). \end{equation} Using the above equation, I get \begin{equation} \frac{d^4}{dx^4}(P_l(x))=\frac{1}{4}\sum_{n=(0,1),2}^{l-2}\sum_{m=(0,1),2}^{n-2}(l-n)(l+n+1)(2n+1)(n-m)(n+m+1)(2m+1)P_m(x). \end{equation} The problem is that I want the double sum in the above as a single sum, just in terms of $n, l$. Is there any idea how to simplify the above?

Answer 1

The $\sum_{n=(0,1),2}$ means that the sum either starts from 0 or 1 depending upon whether or not $l$ is even, and the $n$ changes its value by step of 2.