Associated Legendre functions and orthogonality

[soikez]

According to the orthogonality property of the associated Legendre function

$$P_l^{|m|}(cos\theta)$$

we have that:

$$\int_{0}^{\pi}P_{l}^{|m|}(cos\theta){\cdot}P_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=\frac{2(l+m)!}{(2l+1)(l-m)!}{\delta}_{ll'}$$

What I am looking for is an orthogonality property for the derivative of the associated Legendre function

$$P^{'}_{l}^{|m|}(cos\theta)$$:

something like that perhaps:

$$\int_{0}^{\pi}P^{'}_{l}^{|m|}(cos\theta){\cdot}P^{'}_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=?$$

or even taking into consideration the fact that the derivative of the associated Legendre function is:

$$P^{'}_{l}^{|m|}(cos\theta)=\frac{lcos{\theta}P_{l}^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)}{sin\theta}$$

after some manipulations on my equation an orthogonality property over the sum below:

$$\sum_{l}^{\infty}\sum_{m=-l}^{m=l}cos{\theta}P_{l}^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)e^{jm\phi}$$

Thanks in advance

 

[colinjohnstoe]

I am looking for a similar thing. I have looked in all the spherical harmonics textbooks I can find, but have had no luck. I will let you know if I have anything.