Optimal discrete Hardy-Rellich-Birman inequalities

We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power $\mathrm{\ell }$ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit optimal discrete Rellich ($\mathrm{\ell }=2$) and Birman ($\mathrm{\ell }\ge 3$) weights. For $\mathrm{\ell }=1$, we rediscover the optimal Hardy weight of Keller-Pinchover-Pogorzelski. For $\mathrm{\ell }=2$, we improve upon the best known Rellich weights due to Gerhat-Krejčiřík-Štampach and Huang-Ye. For $\mathrm{\ell }\ge 3$, our main result proves a conjecture by Gerhat-Krejčiřík-Štampach and improves the discrete analogue of the classical Birman weight due to Huang-Ye to the optimal.