  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.
