One-Dimensional Discrete Hardy and Rellich Inequalities on Integers

Abstract

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form $$n^\alpha $$ n α . We prove the inequality when $$\alpha $$ α 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 α ) which are asymptotically sharp as $$\alpha \rightarrow \infty $$ α → ∞ . 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.