I'd like to calculate the convolution of associated Legendre functions of different degrees, i.e. the following integral: $\int_0^\pi P^m_l(\cos\theta)P^m_{l'}(\cos(\theta+\tau))\sin\theta d\theta$
The closest I could get is this: $\int^\pi_0 P_l(\cos\Delta)\sin\theta d\theta=\sum_{m=0}^l \int_0^\pi P^m_l(\cos\theta)P^m_l(\cos\theta')\sin\theta d\theta$. But I don't know how to go from this to the solution I'm interested in. I suppose this is not the right way to go.
Any help is welcome. Thank you in advance.