I come across the following infinite series. 


$$
\sum\_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$


In particular, I am interested in the case where $a=1/4$.


Thanks for any hints and references!


Anand



Has anyone seen this series?
For natural values of *a*, take $\displaystyle\frac{e^t-1}t=\sum\_1^\infty\dfrac{t^{n-1}}{n!}$, then apply the operator $\bigg(\displaystyle\frac1t\cdot\int\bigg)$ *a* times to it. For $a\not\in\mathbb N$, such as $a=\dfrac14$, welcome to the “marvelous” world of [fractional calculus](http://en.wikipedia.org/wiki/Fractional_calculus) and [Riemann-Liouville integrals](http://en.wikipedia.org/wiki/Riemann-Liouville_integral).




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For natural values of *a*, the series can also be expressed in terms of the ([generalized](http://en.wikipedia.org/wiki/Generalized_hypergeometric_function)) [hypergeometric function](http://en.wikipedia.org/wiki/Hypergeometric_function) $\_{a+1}{\large F}\_{a+1}\bigg(\underbrace{1,1,\ldots,1}\_{a+1}~;~\underbrace{2,2,\ldots,2}\_{a+1}~;~t\bigg)$.


