How to compute this limit involving the associated Legendre function?

Question

I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the behavior of the following function, $$f(\theta) = \frac{P^\mu_{\nu}(\cos \theta)}{\sin^\mu \theta}$$ when $\theta\to 0$ and $\theta \to \pi,$ where $\mu,\nu\in \mathbb{R}$ and $\nu$ is a parameter depending on the eigenvalue.

According to the DLMF entry here, we see that $P^\mu_{\nu}(\cos \theta) \sim_{\theta \to 0} c_{\mu}(1-\cos \theta)^{-\mu/2}$ and so $$f(\theta)\sim_{\theta \to 0} c_{\mu} (1-\cos \theta)^{-\mu}$$ when $\mu \neq 1,2,\cdots.$ This is not quite helpful since I would like to obtain some condition on $\nu$ (which will tell me about the eigenvalues) based on how $f$ behaves at the boundaries, i.e. $\theta \in \{0,\pi\}.$ So, I was wondering if there are any asymptotic expressions for the function $f$ in terms of the parameter $\nu$ as $\theta\to 0$ or $\theta \to \pi?$

Answer 1

I find it useful to represent the Legendre function in terms of a hypergeometric function, using a formula from Wikipedia,

$$f(\theta)=\frac{ (1+\cos \theta)^{\mu/2} \, _2F_1\left(-\nu,\nu+1;1-\mu;\frac{1}{2} (1-\cos \theta)\right)}{\Gamma (1-\mu)\sin^\mu\theta(1-\cos \theta)^{\mu/2}}.$$

Then Mathematica gives me the small-$\theta$ expansion $$f(\theta)=\frac{ 4 (1-\mu)+\theta^2 \bigl(\tfrac{1}{3}(1-\mu) \mu- \nu (\nu+1)\bigr)+{\cal O}(\theta^4)}{2^{2-\mu}\theta^{2\mu} \Gamma (2-\mu)}.$$ The leading order term is $\nu$-independent, as noted in the OP, the next order term does depend on $\nu$.