Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$.
I am wondering if one can get nice representation of $L^2$-norm of the function $f^{\alpha}(x)$, namely
$$
\int_{-\infty}^{\infty}(f^{\alpha}_+(x))^2dx.
$$
(Here $y^{\alpha}_+=\max(0, y)^{\alpha}$ is a one-sided power function).
Thank you.
