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 \(n^\alpha \). We prove the inequality when \(\alpha \) 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 \(n^\alpha \)) which are asymptotically sharp as \(\alpha \rightarrow \infty \). 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.