18 problems

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 $\mathcal P(n)$ 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…

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…

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…

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 $k$…

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…

Billey–Burdzy–Sagan's positivity conjecture. Each coefficient $(\Delta^j p_S)(m)$ is a positive integer for $1\le j\le m-1$.

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

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…

Let $\{\alpha_i\}_{i=0}^\infty$ be a non-trivial sequence, meaning it is not zero for all but at most two indices, and suppose that $\alpha_i>0$ for some $i$. A discrete multiplier…

Let $U_T=u(T,\cdot)$ be the solution of the initial-boundary value problem at time $T$ for a given Brownian path, and let $\widehat{U}_T$ be its numerical approximation with spatia…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, let $\mu(p)$ be the mesh size, and let $m$ denote Lebesgue measure. Discrete…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, and let $\mu(p)$ be the minimum spacing between neighboring zeros. For $p$, d…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, let $\mu(p)$ be their mesh size, and define … Rescaled discrete Laguerre conj…

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros. For such a polynomial $p$, let $\mu(p)$ be the minimum spacing between neighb…