How can one compute the *Neumann spectral fractional Laplacian* of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space. We recall 
$$(-\Delta)^\alpha u: \sum_{k=1}^\infty \sqrt 2 c_n\cos(\pi n x) \mapsto  \sum_{k=1}^\infty \sqrt 2 (\pi n)^{2s} c_n\cos(\pi n x),$$
 where $c_n = \int_{0}^{1} \cos(\pi n x) u(x) dx$.

Note that in https://mathoverflow.net/questions/361043/computing-the-fractional-laplacian-of-power-function the formula is given in $\mathbb R^n$ instead of in the interval $(0,1)$.