Optimal fractional discrete Hardy inequalities on the half-line

We consider a Toeplitz realisation of the fractional discrete Laplacian $(-Δ)^α$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to $\ell^{2}(\mathbb{N})$. For all $α>0$, we prove that the fractional Hardy inequality $$(-Δ)^α\geq\frac{4^αΓ^2(α+1/2)}π\frac{Γ(2\,\cdot\,-1)}{Γ(2\,\cdot\,-1+2α)}$$ holds on $\ell^{2}(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of $\ell^{2}(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.