One-Dimensional Discrete Hardy and Rellich Inequalities on Integers

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form \(n^\alpha \) $n^{\alpha }$ . We prove the inequality when \(\alpha \) $\alpha$ is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities(with weights \(n^\alpha \) $n^{\alpha }$ ) which are asymptotically sharp as \(\alpha \rightarrow \infty \) $\alpha \to \infty$ . As a by-product of this work we derive a combinatorial identity using purely analytic methods, which suggests a plausible correlation between combinatorial and functional identities.