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(x) which correspond to that both of the polynomials in the symmetric decomposition of f(x) 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.

中文翻译：

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由 马、Qi、Yeh 和 Yeh 引起的 Boros-Moll 多项式的猜想


伽马正性是具有对称系数的多项式可能具有的基本性质之一，这直接意味着它们是单峰的。它起源于 Foata 和 Schützenberger 对欧拉多项式的研究。然后，对称多项式的交替 gamma 正性由 Sagan 和 Tilrell 定义。后来，马等人进一步引入了多项式f（x）的双伽马正和交替双伽马正的概念，这对应于f（x）对称分解中的两项式分别是伽马正和交替伽马正。在本文中，我们通过验证它们的倒数对称分解中的两个多项式是单峰的和交替的 gamma 正性，建立了 Boros-Moll 多项式的交替双 gamma 正性。它证实了马、齐、叶和叶提出的一个猜想。