Express an exponential integral in hypergeometric form

Question

I am new to hypergeometric function. I am trying to express this:

$$\int_{0}^{\infty}e^{-ax^k+bx}dx$$

in a hypergeometric form. I have read some reference, but I don't get it how to cope with this one. thanks for your help!!

Answer 1

I tried to expand $e^{-ax^b}$ first, and finally I get an answer like:

$\sum\limits_{n=0}^{\infty}\frac{-a^n}{n!} \left(b\right)^{-nk-1}\Gamma(nk+1)$.

this answer is ok for k <1.