Peaks and Gaussian Limits for Powered Binomial-Sum Ratios
Overview
Ratios of truncated binomial power sums to full sums in shorter rows: the conjectured maximum asymptotic, shifted integer peaks and a Gaussian limiting profile for fixed positive powers and weights, with strict log-concavity for the square power.
Original public questions
Byun–Poznanović Conjecture 1.1(d): generalized Glasby–Paseman ratios
Seok Hyun Byun and Svetlana Poznanović: Unimodality and log-concavity of generalized Glasby-Paseman sequences — arXiv 2604.14639v1, page 2: Conjecture 1.1(d), the asymptotic maximum of the generalized Glasby-Paseman ratio for every fixed source power and weight. The full four-part Conjecture 1.1 is not claimed solved.
Original abstract (English)
We determine the asymptotic maximum of the ratio between a truncated powered binomial sum and the corresponding full sum in a shorter row. This proves the maximum-value conjecture of Byun and Poznanović for every fixed positive integer power and positive weight, and extends it to all positive real powers. Every maximizing index lies within one half, up to an error of order n^-1, of an explicitly shifted linear function of the row length. We also determine the first lattice correction to the maximum. After normalization, the entire sequence satisfies a uniform local Gaussian limit, with convergence of all fixed absolute centered moments and an explicit constant correction to the mean. For the square power, we prove strict log-concavity for every positive weight and every row length. The asymptotic proof combines a geometric endpoint estimate, a Gaussian denominator estimate and global entropy localization.
Mathematical review
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