31 problems

Let $m\in\mathbb{N}$ and let $x$ be a positive integer. The coefficients ${\mathbf{A}}_{m,0}, {\mathbf{A}}_{m,1},\ldots,{\mathbf{A}}_{m,m}$ are constants independent of $x$ and $k$…

Threshold conjecture. For $1\leq m\leq 11$ and $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the Laguerre inequality of order $m$ whenever…

Billey–Burdzy–Sagan peak polynomial positivity conjecture. For every admissible set $S$ with $m=\max(S)$, each coefficient $(\Delta^j p_S)(m)$ is a positive integer for…

The main conjecture. The Whitney constant satisfies $W_m=1$.

Let $p$ be a real-rooted polynomial of degree $k$, and let $M(p)$ denote its mesh, with $M(p)\geq 1$. Discrete Laguerre inequality. Then … This conjecture proposes a discrete analo…

Let $p(n)$ denote the partition function, let $\Delta$ be the forward difference operator $\Delta(a)(n)=a(n)-a(n-1)$, and let $n_0(r)$ be a threshold such that $\Delta^r(p)(n)>0$ f…

A Riemann difference of order $n$ is determined by its nodes and coefficients; it has the Marcinkiewicz-Zygmund (MZ) property when … imply $f(h)=o(h^n)$, where $Rf(h)$ denotes the…

Let $\mathcal P(n)$ be the set of partitions of $n$, let $\mathcal P[m,n]=\biguplus_{i=m}^n\mathcal P(i)$, and let…

Threshold conjecture. For $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the third-order Turán inequality whenever $n\geq T_p(k)$, respectively…

GGR conjecture. The $\binom{n+1}{2}$ limits $D_{m,k}$ for $0\leq k<m\leq n$ should be enough for $f$ to be $n$ times Peano differentiable at $c$.

Let $\hat{p}_i(u)=\sum_{k=0}^5 a_k (u-d_i^L)^k$ be the quintic piece joining the cubic pieces $p_i$ and $p_{i+1}$ with matching values, first derivatives, and second derivatives at…

Finite-difference sign conjecture. For any $k\geq 1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that, for $n\geq n_k(r)$,

Let $l$ be a positive integer, and let $P(x)$ be a real polynomial of degree $n$ with leading coefficient $a_n$. Scaled generalized Tepper identity. One has … The preceding theorem…

Alternating Pascal-row identity. This inner product is equal to $n!$.

Let $\mathbb{R}_+=(0,\infty)$, let $\mathcal{K}^{\infty}$ denote the class of functions that are completely convex, and let $\mathcal{D}^{\infty}_{\mathbb{N}}$ denote the class of…

Weighted Catalan 2-adic valuation conjecture. If $b$ satisfies

Let $r\in\mathbb{N}$, $v>0$, $m\in\mathbb{N}\cup\\{0\\}$, and $0<\alpha<1$. Define $P_r(\alpha,m,v)$ as the polynomial appearing in the identity below. Polynomial remainder conject…

The paper considers the generalized factorial functions and the associated formulas for the sequences $p_n(\alpha,R)$, which have been proved in the special cases…

Degree conjecture. One has $\operatorname{deg}_{t}H_{m,n}(X,t)=m-1$. In particular,

Let $p(n)$ denote the number of partitions of the nonnegative integer $n$, and let $\Delta$ be the difference operator with respect to $n$. For a fixed integer $r$, let $n(r)$ be a…

Let $p(n)$ denote the number of partitions of the nonnegative integer $n$, and let $\Delta$ be the difference operator with respect to $n$. For a fixed integer $r\geq 1$, write…

Let $p(n)$ denote the number of partitions of the positive integer $n$. For integers $n\geq 45$, consider the consecutive ratio inequality involving $p(n-1)$, $p(n)$, and $p(n+1)$.…

Let $p(x)$ be a hyperbolic polynomial, and let $\lambda$, $\delta$, and $\gamma$ be non-negative real numbers with $\lambda\leq\delta$. Assume that … For the finite-difference func…

For polynomials of degree at most $d$, define the forward difference operator by … and define the product … Here a hyperbolic polynomial has mesh at least one when consecutive root…

Let $T$ be a constant-coefficient difference operator … It acts on hyperbolic polynomials of degree at most $m$ whose mesh is at least one, where the mesh is the minimum distance b…