I am trying to estimate the following expectation value in the multinomial probability distribution:


\begin{equation}
\mathbb{E}\_{P}\left[ \left( \frac{n!}{x\_1!..x\_k!}\right)^{\alpha - 1} \right]
\end{equation}


where $P$ the usual multinomial distribution, $\theta\_i \in [0,1] \forall i$, $\sum\_{i=1}^{k} \theta\_i = 1$, $\alpha \in (0,+\infty)\setminus \{1\}$ and $\sum\_{i=1}^k x\_i = n$.


1) Thus, I am wondering if there exists an extension of the multinomial theorem for the following:


\begin{equation}
\sum\_{x\_1 + .. + x\_k = n} \left( \frac{n!}{x\_1!..x\_k!}\right)^{\alpha} \theta\_1^{x\_1}..\theta\_k^{x\_k}
\end{equation}


2) If not, would there be a way to estimate this w/o using Stirling on the $x\_i$?


I found asymptotic estimates of the sum of powers of multinomial coefficients in my search (<https://arxiv.org/abs/0807.5028>), however this does not seem very useful here.


Any help would be greatly appreciated. 


Thank you in advance



Generalization of multinomial theorem for powers of multinomial coefficients
I doubt there exists an closed-form expression for 
\begin{equation}
S\_\alpha := \sum\_{x\_1 + .. + x\_k = n} \left( \frac{n!}{x\_1!..x\_k!}\right)^{\alpha} \theta\_1^{x\_1}..\theta\_k^{x\_k}.
\end{equation}
Using the [power mean inequality](https://en.wikipedia.org/wiki/Generalized_mean#Generalized_mean_inequality), it can be bounded as follows:
$$S\_\alpha \lessgtr \binom{n+k-1}{k-1}^{1-\alpha}(\theta\_1^{1/\alpha}+\cdots+ \theta\_k^{1/\alpha})^{n\alpha},$$
where the sign is $\geq$ when $\alpha>1$, and $\leq$ otherwise.


