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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?

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  • $\begingroup$ If the series converges absolutely, you can use any reasonable theorem. Of course, you should interpret the coefficients correctly. $\endgroup$ – Alex Degtyarev Feb 2 '14 at 9:29
  • $\begingroup$ @Alex, ok... But I don't know if it applies to this theorem specifically (or if it works only for a series of simple real numbers): en.m.wikipedia.org/wiki/Multinomial_theorem $\endgroup$ – Rafael Feb 2 '14 at 17:34
  • $\begingroup$ As I said: you can use the theorem if you know that the convergence is absolute. $\endgroup$ – Alex Degtyarev Feb 2 '14 at 17:39

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