On the gamma-positivity of the Eulerian transformation

Christos A. Athanasiadis

Published 2023-02-01Version 1

The Eulerian transformation is the linear operator on polynomials in one variable with real coefficients which maps the powers of this variable to the corresponding Eulerian polynomials. Br\"and\'en and Jochemko have shown that the Eulerian transforms of a class of polynomials with nonnegative coefficients, which includes those having all their roots in the interval $[-1,0]$, have nonnegative and unimodal symmetric decompositions. The stronger statement that these symmetric decompositions are $\gamma$-positive is proven in this paper and is generalized in the context of uniform triangulations of simplicial complexes. The real-rootedness of these generalized transformations is also discussed.