# Product of two spherical harmonics

According to wikipedia, the product of two spherical harmonics is

this equation

What I do not understand is the scope of the sum. From where to where does it go, over all integers for L and M?

Also since $m1 + m2 + M$ has to be zero for the Wigner symbols to be non-zero, couldn't we simply insert $M = -m1 - m2$ into the sum?

In my case $|m_1| \leq l_1$ and $|m_2| \leq L_2$ with all being integers.

The sum $\sum_{LM}$ should be understood as $\sum_{L=0}^{\infty} \sum_{M=-L}^{L}$.
For the second question, that's a good point... I am not completely sure but it does look indeed as if you could replace $M$ by $m_1+m_2$. So the sum over $M$ is killed, then... That's surprising.