In the context of Hardy inequalities for the fractional Laplacian \((-\Delta _{\mathbb {N}})^{\sigma }\) on the discrete half-line \(\mathbb {N}\) , we provide an optimal Hardy-weight \(W^{\textrm{op}}_{\sigma }\) for exponents \(\sigma \in \left( 0,1\right] \) . As a consequence, we provide an estimate of the sharp constant in the fractional Hardy inequality with the classical Hardy-weight \(n^{-2\sigma }\) on \(\mathbb {N}\) . It turns out that for \(\sigma =1\) the Hardy-weight \(W^{\textrm{op}}_{1}\) is pointwise larger than the optimal Hardy-weight obtained by Keller–Pinchover–Pogorzelski near infinity. As an application of our main result, we obtain unique continuation results at infinity for the solutions of some fractional Schrödinger equation.
