In order to carry out an integral, I'm using a series definition of associated Laguerre polynomials:
$$ L_\ell ^ \alpha (x) = \sum_{\mu = 0}^{\ell} \left( {\begin{array}{} \ell + \alpha \\ \ell - \mu \\ \end{array}} \right) \frac{(-x)^\mu}{\mu !} $$
Since I need to square it but want to maintain its summation form, is this correct?
$$ \left[ L_\ell^\alpha (x) \right ]^2 = \sum_\nu^\ell \sum_\mu^\ell \left( {\begin{array}{*{10}c} \ell + \alpha \\ \ell - \mu \\ \end{array}} \right) \left( {\begin{array}{*{10}c} \ell + \alpha \\ \ell - \nu \\ \end{array}} \right) \frac{(-x)^{\mu+\nu}}{\mu!\nu!} $$
As a general rule, if you have the sum $$\bigg(\sum_{i=1}^m a_i\bigg)\bigg(\sum_{j=1}^n b_j\bigg)$$ This product of sums can be transformed into the double sum $$\sum_{i=1}^m \sum_{j=1}^n a_i b_j$$ and thus the square of a sum $$\bigg(\sum_{i=1}^m a_i\bigg)^2$$ is equal to $$\sum_{i=1}^m \sum_{j=1}^m a_i a_j$$ So yes, what you've done should be correct.