I have been searching for different solutions for the derivatives of associated Legendre polynomials at the end points. The associated Legendre polynomial is defined as: $$P^m_l(x)=(-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}P_n(x)$$ See matlab's link
The derivative of the associated Legendre polynomials can be defined using a recurrence relationship where the derivative is defined by other polynomials within the associated Legendre polynomial family. In this case I am trying to use the recurrence described on wikipedia's page:
$$(x^2-1)\frac{d}{dx}P_l^m(x)=\sqrt{1-x^2}P_l^{m+1}(x)+mxP_l^m(x)$$
To solve for the derivative divide both sides by $(x^2-1)$ and you will see the function goes unbounded at $x= \pm 1$ which are the end points of the function also it can be undefined (i.e., $0/0$) for when $m = 0$ and $x=\pm 1$. How can I compute the derivative at these end points? L'Hopital's rule does not work here. I'm wondering If i can use something like a Laurent series or some kind of series to approximate it but with the end goal of implementing this into code eventually.
As I continue my search for solutions has anyone had experience in computing the endpoints of the first derivative of the associated Legendre polynomials?