What are the conditions for using the multinomial theorem with infinite series? I have an expression but I don't know if I can use it. The expression is:

$$ \left[\sum_{m=0}^{\infty} \frac{\mu^m \kappa^m}{m! \Gamma\left(\mu+m\right)} \alpha^{-(\mu+m)} \Gamma\left(\mu+m, \beta\right)\right]^L $$

in which $\Gamma(\cdot, \cdot)$ is the incomplete gamma function.

Can I use the theorem? This allows me to transform the power in a simple summation?