A conjecture on Boros-Moll polynomials due to Ma, Qi, Yeh and Yeh

Abstract

Gamma-positivity is one of the basic properties that may be possessed by polynomials with symmetric coefficients, which directly implies that they are unimodal. It originates from the study of Eulerian polynomials by Foata and Schützenberger. Then, the alternatingly gamma-positivity for symmetric polynomials was defined by Sagan and Tirrell. Later, Ma et al. further introduced the notions of bi-gamma-positive and alternatingly bi-gamma-positive for a polynomial $f\left(x\right)$ which correspond to that both of the polynomials in the symmetric decomposition of $f\left(x\right)$ are gamma-positive and alternatingly gamma-positive, respectively. In this paper we establish the alternatingly bi-gamma-positivity of the Boros–Moll polynomials by verifying both polynomials in the symmetric decomposition of their reciprocals are unimodal and alternatingly gamma-positive. It confirms a conjecture proposed by Ma, Qi, Yeh and Yeh.