Related papers: Log-Concavity of Infinite Product Generating Funct…

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…

D. Uminsky and K. Yeats [6] studied the properties of the log- operator L on the subset of the finite symmetric sequences and prove the existence of an infinite region R, bounded by parametrically de- fined hypersurfaces such that any…

The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo 2. Our…

For $m,n\in \mathbb{N}$, let $0 < \alpha_i,\beta_j,\lambda_{ij} \leq 1$ be such that $\sum_{j=1}^n \lambda_{ij} = \alpha_i$, $\sum_{i=1}^m \lambda_{ij} = \beta_j$, and $\sum_{i=1}^m \alpha_i = \sum_{j=1}^n \beta_j \leq 1$. We prove that the…

In his important 1920 paper on partitions, MacMahon defined the partition generating functions \begin{align*} A_k(q)=\sum_{n=1}^{\infty}\mathfrak{m}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k}…

Let $p$ be prime, and let $p_{[1,p]}(n)$ denote the function whose generating function is $\prod (1-q^n)^{-1}(1 - q^{pn})^{-1}$. This function and its generalizations $p_{[c^{\ell}, d^m]}(n)$ are the subject of study in several recent…

In this paper we provide a classification on the sign distribution of $\Delta _{E,\ell}(n):= p_{E,\ell }(n)^2 - p_{E,\ell }(n-1) \, p_{E,\ell }(n+1)$, where \begin{equation*} \sum_{n =0}^{\infty} p_{E,\ell }(n) \, q^n := \prod_{n \in S}…

We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\bf Strongly Log-Concave} entire functions, generalizing {\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$…

Lin introduced the partition function $\text{PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of $n$ with designated summands in which all parts are odd. For $k\geq0$, Lin conjectured congruences for…

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum…

Almost nothing is known about the parity of the partition function $p(n)$, which is conjectured to be random. Despite this expectation, Ono surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4…

Bessenrodt and Ono initially found the strict log-subadditivity of partition function $p(n)$, that is, $p(a+b)< p(a)p(b)$ for $a,b>1$ and $a+b>9$. Many other important statistics of partitions are proved to enjoy similar properties. Lovejoy…

We investigate the convexity property on $(0,1)$ of the function $$f_a(x)=\frac{{\cal K}{(\sqrt x)}}{a-(1/2)\log(1-x)}.$$ We show that $f_a$ is strictly convex on $(0,1)$ if and only if $a\geq a_c$ and $1/f_a$ is strictly convex on $(0,1)$…

Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher order $\log$-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This…

Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic…

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms $a\mapsto\sum{f_k}(a)_kx^k$, $a\mapsto\sum{f_k}\Gamma(a+k)x^k$ and $a\mapsto\sum{f_k}x^k/(a)_k$. The most useful examples of such functions are…

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence…

Let $d\in\mathbb N$ and $f$ be a function in the Orlicz class $L(\log^+L)^{d-1}$ defined on the unit cube $[0,1]^d$ in $\mathbb{R}^d$. Given partitions $\Delta_1,\ldots,$ $\Delta_d$ of $[0,1]$, we first prove that the orthogonal projection…

In recent years, the log-concavity of $\{\sqrt[n]{S_n}\}_{n\geq 1}$ have been received a lot of attention. Very recently, Sun posed the following conjecture in his new book: the sequences $\{\sqrt[n]{a_n}\}_{n\geq 2}$ and $\{…

Let $G_n= \prod_{k=0}^n \binom{n}{k},$ the product of the elements of the $n$-th row of Pascal's triangle. This paper studies the partial factorizations of $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having $p \le…