One dimensional sharp discrete Hardy-Rellich inequalities

In this paper, we establish discrete Hardy-Rellich inequalities on $\mathbb{N}$ with $\Delta^\frac{\ell}{2}$ and optimal constants, for any $\ell \geq 1$. As far as we are aware, these sharp inequalities are new for $\ell \geq 3$. Our approach is to use weighted equalities to get some sharp Hardy inequalities using shifting weights, then to settle the higher order cases by iteration. We provide also a new Hardy-Leray type inequality on $\mathbb{N}$ with the same constant as the continuous setting. Furthermore, the main ideas work also for general graphs or the $\ell^p$ setting.