Positive Expansions of Optimal Discrete p-Birman Weights

We prove the positive-coefficient expansion conjectured in Remark 2.14 of Štampach and Waclawek,
Optimal discrete p-Hardy–Rellich–Birman inequalities, arXiv:2605.25238v1. For every integer ℓ ≥ 1
and every real p > 1, all correction coefficients in the expansion of their gamma-ratio weight in
negative powers of n are strictly positive. The expansion converges at every n ≥ ℓ. The proof reduces
the weight to a finite difference of products and establishes coefficientwise positivity by a discrete
product rule. We also compute the first correction coefficient and obtain an explicit strengthened
lower bound and a monotone sequence of truncated Hardy–Rellich–Birman weights.