# About one series. Are there some related special functions?

Hello,

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! :-)

Anand

-