I am working with spherical harmonics and the radial equation (part of Laplace's equation in spherical co-ords). The coefficients and equations with I am working with aren't important to my question.
I am using the orthogonality relation for spherical harmonics:
$<Y_{kn},Y_{lm}> = \int^{2\pi}_0\int^{\pi}_0 Y^*_{lm} Y_{kn} sin(\theta) d\theta d\phi = \delta_{kl} \delta_{mn}$
In the problem below:
$<Y_{kn},f> = \sum\limits_{l=0}^\infty\sum\limits_{m=-l}^\infty (A_{lm}a^l)<Y_{kn},Y_{lm}>$
We can use the orthogonality relation to get:
$<Y_{kn},f> = \sum\limits_{l=0}^\infty\sum\limits_{m=-l}^\infty (A_{lm}a^l)\delta_{kl} \delta_{mn}$
Which simplifies to:
$<Y_{kn},f> = A_{kn}a^k$
Why do the Kronecker deltas "get rid off" the summation signs and why have we relabelled the coefficients on the right hand side?
Because
$$\delta_{kl}=\begin{cases}0&,\;\;k\neq l\\{}\\1&,\;\;k=l\end{cases}$$
so in that sum all equals zero except when $\;l=k\;,\;m=n\;$