I have the following series:

$$ \sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0, $$

where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum gives $t(e^t-1)$. When $a=1/2$, it gives $e^{t^2}t^2 (1+ Erf(t))$ where $Erf(\cdot)$ is the standard error function. It is not difficult to see that it indeed converges. For general $a\in (0,1]$, are there some special functions related to this sum?

Thank you very much for any hints and helps! :-)



It's a special case of the Mittag-Leffler function.


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