Reflected Newton Transforms and Concavity of Peak-Polynomial Coefficients

We study a polynomial transform defined by reflected backward differences. We use the backward differences at a fixed center as its input coordinates, called Newton data. In these coordinates, its rising-binomial coefficients are obtained from an explicit matrix with symmetric, tent-shaped columns. This formula determines the kernel of the transform and shows that its image on nonnegative Newton data is exactly the cone of symmetric sequences that remain concave after zero padding at the two endpoints. Every integral sequence in this cone has an explicit preimage with integral Newton data. The same representation gives strong coefficientwise log-concavity for arbitrary multivariate nonnegative Newton data, both in the rising basis and in forward binomial bases. We also characterize when the recurrence starts with the zero polynomial and prove that the two-unit shift of the difference center required by the positivity criterion is sharp. As applications, peak-polynomial coefficients in Bencs's rising basis are positive integers, symmetric and ordinarily concave, proving his log-concavity conjecture. We determine their exact maximum plateau and all log-concavity equality cases. The forward-basis result answers a question of Diaz-Lopez, Harris, Insko and Omar at every integer center at or beyond the last peak.