Higher Tur\'{a}n inequalities for the plane partition function

Abstract

Here we study the roots of the doubly infinite family of Jensen polynomials $J_{\mathrm{PL}}^{d,n}(x)$ associated to MacMahon's plane partition function $\mathrm{PL}(n)$. Recently, Ono, Pujahari, and Rolen proved that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12$, which is equivalent to the polynomials $J_{\mathrm{PL}}^{2,n}(x)$ having real roots. Moreover, they proved, for each $d\geq 2$, that the $J_{\mathrm{PL}}^{d,n}(x)$ have all real roots for sufficiently large $n$. Here we make their result effective. Namely, if $N_{\mathrm{PL}}(d)$ is the minimal integer such that $J_{\mathrm{PL}}^{d,n}(x)$ has all real roots for all $n\geq N_{\mathrm{PL}}(d)$, then we show that $$N_{\mathrm{PL}}(d)\leq 279928\cdot d(d-1)\cdot \left(6 d^3\cdot (22.2)^{\frac{3(d-1)}{2}}\right)^{2d} e^{\frac{\Gamma(2d^2)}{(2\pi)^{2d+2}}} .$$ Moreover, using the ideas that led to the above inequality, we explicitly prove that $N_{\mathrm{PL}}(3)=26, N_{\mathrm{PL}}(4)=46, N_{\mathrm{PL}}(5)=73, N_{\mathrm{PL}}(6)=102$ and $N_{\mathrm{PL}}(7)=136$.