与Wright圆法相关的Jensen多项式：双曲性与Turán不等式

Jensen--Pólya 纲领与算术序列的不等式

The Jensen--Pólya program and inequalities for arithmetic sequences

本文建立解析框架研究算术序列的 Jensen 多项式双曲性及相关不等式，给出 Hermite--Jensen 现象的部分逆命题，并证明分拆型函数的多类不等式猜想。

A note on the Higher order Turán inequalities for $k$-regular partitions

Nicolas and DeSalvo and Pak proved that the partition function $p(n)$ is log concave for $n \geq 25$. Chen, Jia and Wang proved that $p(n)$ satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for $n \geq 94$. More recently, Griffin, Ono, Rolen and Zagier proved more generally that for all $d$, the degree $d$ Jensen polynomials associated to $p(n)$ are hyperbolic for sufficiently large $n$. In this paper, we prove that the same result holds for the $k$-regular partition function $p_k(n)$ for $k \geq 2$. In particular, for any positive i…

Hermite-Jensen极限与$q$-多项式$ d $ log-concavity

Hermite-Jensen limits and $d$ log-concavity of $q$-multinomials

研究$q$-多项式$ d $ log-concavity的性质，通过Hermite-Jensen极限和归一化Jensen多项式渐进行为，证明在特定条件下，$ q $-多项式满足更强的log-concavity不等式。

Higher Turán inequalities for the plane partition function

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}}…

$(-1)^{r} Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ 的渐近增长与 $\sqrt[n]{\overline{p}(n)/n^α}$ 的反向高阶 Turán 不等式

Asymptotic Growth of $(-1)^{r} Δ^r \log \sqrt[n]{\overline{p}(n)/n^α}$ and the Reverse Higher Order Turán Inequalities for $\sqrt[n]{\overline{p}(n)/n^α}$

本文研究了overpartition函数 $\sqrt[n]{\overline{p}(n)/n^α}$ 对数有限差分的渐近增长，给出了对称上下界不等式，并引入并证明了反向高阶 Turán 不等式，从而刻画了相关 Jensen 多项式的非实根性。

Higher Order Turán Inequalities for the Partition Function

The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence $\{a_{n}\}$ is said to satisfy the Turán inequalities if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Turán inequalities if for $n\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. A sequence satisfying the Turán inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequ…