One-dimensional discrete Hardy and Rellich inequalities on integers

TL;DR

This paper establishes sharp weighted Hardy and Rellich inequalities on integers, providing explicit constants and higher order versions, and uncovers a link between combinatorial and functional identities.

Contribution

It introduces sharp weighted inequalities with explicit constants and higher order versions, and reveals a novel connection between combinatorial and functional identities.

Findings

Proved sharp weighted Hardy inequalities with power weights.

Derived explicit constants for Rellich inequalities and their higher order forms.

Discovered a combinatorial identity linked to the inequalities.

Abstract

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 and its higher order versions. As a by-product of this work we derive a combinatorial identity using purely analytic methods. This suggests a correlation between combinatorial identities and functional identities.