A "generalized" exponential power series

Question

I'm wondering if

$$ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} $$

what would this be

$$ \sum_{k=0}^\infty \frac{x^{k+\alpha}}{\Gamma(k+\alpha)} = \large{?}_{\alpha}(x) $$

for $\alpha \in (0,1)$?

What is the the name of this series? Is it known for particular values of $\alpha$, for example $\alpha=1/2$?

Answer 1

$$\sum_{k=0}^\infty\frac{x^{k+a}}{(k+a)!}~=~e^x~\bigg[1-\dfrac{\Gamma(a,x)}{\Gamma(a)}\bigg].$$

Answer 2

I agree with Lucian and Alex, such a function can be expressed in terms of an incomplete Gamma function or a hypergeometric function $\phantom{}_{1} F_1$. I just wanted to add that for $\alpha=\frac{1}{2}$ we have something that depends on the error function:

$$ \sum_{k\geq 0}\frac{x^{k+1/2}}{(k+1/2)!} = e^x\operatorname{Erf}(\sqrt{x})=\frac{e^x}{\sqrt{\pi}} \int_{-\sqrt{x}}^{\sqrt{x}}e^{-y^2}\,dy.$$