Let $\nu>0$ be fixed. I am interested in the asymptotic behaviour of the series \begin{equation*}s(n,\nu)=\sum_{k=0}^\infty \frac{n^k}{(k!)^\nu} \end{equation*} in the limit $n\rightarrow\infty$. For integer $\nu\geq2$, it was shown by Shmueli et al. 2005 that $s(n,\nu)$ can be represented as the multiple integral \begin{equation*}s(n,\nu)=\frac{1}{(2\pi)^{\nu-1}}\int_{-\pi}^{\pi}\!\cdots\!\int_{-\pi}^{\pi}\exp\bigg\{n^{1/\nu}\bigg(\sum_{k=1}^{\nu-1}\exp(ib_k)+\exp\bigg(-\sum_{k=1}^{\nu-1}ib_k\bigg)\bigg)\bigg\}\,db_1\ldots\,db_{v-1}, \end{equation*} where $i=\sqrt{-1}$. A standard application of Laplace's approximation for multiple integrals then yields the asymptotic formula \begin{equation*}s(n,\nu)\sim \frac{\exp(\nu n^{1/\nu})}{n^{(\nu-1)/2\nu}(2\pi)^{(\nu-1)/2}\sqrt{\nu}}(1+O(n^{-1/\nu})) \quad \mbox{as}\: n\rightarrow\infty, \end{equation*} where $\nu\geq2$ is an integer.
I would like to obtain a similar asymptotic formula for $s(n,\nu)$ for any positive real $\nu$. I have reasons to believe that the above asymptotic formula is valid for all $\nu>0$, although I can't think of an approach that would yield such a result. Any help would be much appreciated.