Real-Rootedness and Gamma-Positivity for a Variation of the Morris Constant Term

Beck and Pixton expressed the Ehrhart polynomial of the Birkhoff polytope as a weighted sum of constant terms of several multivariate rational functions. Xin and Zhang studied a class of constant terms $h_n(t)$, which can be regarded as a variation of the Morris constant term. They proved that $h_n(t)$ is a polynomial of degree $(n-1{)}^2$ and obtained many nice properties involving the Morris constant term identity. Let $h_n^{\ast }(y)=(1-y{)}^{(n-1{)}^2+1}{\sum }_{t\ge 0}{h}_n(t)y^t$. For fixed $n\ge 3$, we obtain the following three main results: (i): $h_n^{\ast }(y)$ is a polynomial with positive integer coefficients. (ii): $h_n^{\ast }(y)$ is real-rooted. In particular, all its roots are non-positive real numbers. (iii): $h_n^{\ast }(y)$ is Gamma-positive. Furthermore, $h_n^{\ast }(y)$ is palindromic, unimodal, and ultra log-concave. This confirms Xin and Zhang’s conjecture regarding $h_n^{\ast }(y)$. As a byproduct, we prove that every root of a Gamma polynomial associated with $h_n^{\ast }(y)$ is a negative real number.