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 Computing the fractional Laplacian of power function the formula is given in $\mathbb R^n$ instead of in the interval $(0,1)$.