Optimal discrete Hardy-Rellich-Birman inequalities

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