I am working on boundary value problems with the associated Legendre Polynomial and have the condition that:
$$\frac{P_n^m(\cos\theta_w)}{\sin(\theta_w)}=0$$
I am trying to solve this equation for $\theta_w$ for any (positive integer) mode numbers ($n,m$).
I started by noting that the denominator is bounded as: $-1<\sin(\theta_w)<1$, and won't contribute to the roots.
This simplifies the solution that I need to solve to:
$$P_n^m(\cos\theta_w)=0.$$
I then began considering the definitions of associated Legendre polynomials (note the chain rule when reparameterizing in terms of angles),
$$P_n^m(\cos\theta_w)=\frac{(-1)^m}{2^nn!}(1-\cos^2\theta_w)^{m/2}\left(\frac{1}{-\sin\theta_w}\right)^{m+n}\frac{d^{n+m}}{d\theta_w^{n+m}}(\cos^2\theta_w-1)^n.$$
Now the factorial terms are constants and don't affect the roots, while the $(1-\cos^2\theta_w)^{m/2}$ gives zeros for $\theta_w=0,\pi,2\pi,3\pi...$, physically this is not realizable for the physical scenario I am modeling so this can be ignored. With the same bounds on $\sin\theta$ from above the $\sin\theta$ term can be ignored too. This reduces the equation that needs to be solved to:
$$0=\frac{d^{n+m}}{d\theta_w^{n+m}}(\cos^2\theta_w-1)^n.$$
Integrating this $m+n$ times gives:
$$(\cos^2\theta-1)^n=\sum_{i=0}^{m+n-1}\frac{c_i\theta_w^{i}}{i!}$$
which is simplified to:
$$-\sin^{2n}\theta_w=\sum_{i=0}^{m+n-1}\frac{c_i(\theta_w)^{i}}{i!}.$$
Here $c_i$'s are constants of integration. I think this is a transcendental equation, and am wondering if there are any analytical solutions known for this equation.
I had also looked at the definitions of associated Legendre polynomials involving hypergeometric functions, but to no avail.
An example for a specific mode with the $m=0,n=2$ gives:
$$0=\frac{P_2^0(\cos\theta_w)}{\sin(\theta_w)}=\frac{(3\cos^2\theta_w-1)}{2\sin(\theta_w)}$$
This gives the solution that $\theta_w=\cos^{-1}(1/9)$. Knowing this root, I found that the transcendental equation would have coefficients $c_1=0,c_0=4/9$, so that it reads $-\sin^{2n}\theta_w=c_0$. I'm sure if this offers any hints at generalizing to any mode numbers ($n,m$).
I can't make a comment to your question because I don't have enough reputation points. Are you trying to model a biconical antenna (or perhaps any conducting conical surface)?
I've recently been looking at that particular problem myself, and I think I've convinced myself that for some arbitrary angle ($\theta_{w}$ in your case) there is no general solution. However, the Associated Legendre Functions might. If you let your order be real-valued ($P_{n}^{m} \rightarrow P_{\nu}^{m}$ for $\nu \in \mathbb{R}$), then you can likely find values of $\nu$ which work (I'm currently on this path myself).