One-Dimensional Discrete Hardy and Rellich Inequalities on Integers

• DOI: 10.1007/s00041-024-10070-6
• Publication Date: 2024-03-08T23:01:26Z
• Authors: Shubham Gupta
• Keywords: Mathematics - Functional Analysis; FOS: Mathematics; 0101 mathematics; 01 natural sciences; Functional Analysis (math.FA)

AbstractIn 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.