$\int_0^1 \mathrm{d}y\:y^n \exp\left[\alpha \sqrt{1-y}\right]$

Question

Let $n$ be a positive integer (or zero) and $\alpha\in\mathbb{R}$, then consider

$$ \begin{equation} \int_0^1 \mathrm{d}y\:y^n \exp\left[\alpha \sqrt{1-y}\right] \end{equation}, $$

We may expand the exponential and then the squared root in power series, but then the expression will be in terms of the remaining sums which is not very useful. I wonder if it's possible to write down an answer in terms of special and/or elemnetary functions.

Answer 1

Hint

$$I_n=\int_0^1y^n e^{\alpha \sqrt{1-y}}\,dy$$ A @tuna commented, let $\sqrt{1-y}=x$ which makes $$I_n=2\int_0^1 x \left(1-x^2\right)^n e^{\alpha x}\,dx$$ Now, use the binomial expansion to face a series of integrals $$J_k=\int_0^1 x^{2k+1}e^{\alpha x}\,dx$$ and think about the incomplete gamma function.