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.