Properties of a Mehler's type integral

Question

When computing the resolvent of the Laplace beltrami opetator on $S^n$ for even dimension, $n=2k$, I came across the following integral $$ F(\theta)=\int_{-\theta}^{\theta}{\frac{e^{(i\lambda-\mu)\phi}}{(2\cos \phi-2\cos \theta)^{1/2}} d\phi}, ~~~~~ \theta\in(0,\pi) $$ and I need to estimate $(\frac{1}{\sin \theta}\frac{\partial}{\partial\theta})^kF(\theta)$. I know that there is a similar formula called Mehler's integral $$ P_n(\cos\theta)=\int_{-\theta}^{\theta}{\frac{\cos (n+\frac{1}{2})\phi}{(2\cos \phi-2\cos \theta)^{1/2}} d\phi} $$ I want to know if there is a similar expression for $F(\theta)$, and do we have a expansion of $F(\theta)$ like $\sum a_k(\theta,\lambda,\mu)e^{(i\lambda-\mu)\theta}$? The reason I ask the second question is that the resolvent in the odd dimension can be written as such forms, so I wonder if it's also the case in even dimention.

Answer 1

There exists a very similar expression in your case. Actually the formula 2.5.16.1 in the first volume of Prudnikov-Brychkov-Marychev gives even a one-parameter generalization of what you need:

$$\int_0^{\theta}\left(\cos\phi-\cos\theta\right)^{\nu-1}\cos b\phi\,d\phi=\sqrt{\frac{\pi}{2}}\Gamma(\nu)\sin^{\nu-\frac12}\theta \;P^{\frac12-\nu}_{-\frac12+b}(\cos\theta),$$

where $P_{\lambda}^{\mu}(x)$ denotes the Legendre function. It remains to set $\nu=\frac12$ in the above.