11 problems

Laguerre threshold conjecture. For $k=1$ or $2$ and $1\leq m\leq14$, the sequence $\Delta_k(n)$ satisfies the Laguerre inequality of order $m$, namely $L_m(\Delta_k(n))\geq0$, for…

Generalized Laguerre conjecture. Given $t>0$, the polynomial $h_n(x;t)$ satisfies

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 $\Delta_k(n)$ be the broken $k$-diamond partition function, and set $a_n=\Delta_k(n)$. Define … … and $I=A^3-27B^2$. Invariant positivity conjecture. For $k=1$ or $2$ and…

Dong–Ji's conjecture. For the sequence $a_n=q(n)$, one has $A>0$ for $n\geq230$, $B>0$ for $n\geq272$, and $I>0$ for $n\geq267$. The source records that the first two thresholds we…

Let $\phi(x)$ belong to the positive shifted Laguerre–Pólya class $\mathcal{SL\text{-}P}^{+}$, and let $L_j$ be the Laguerre operator defined by the coefficient expansion … The Lag…

Special-case generalized Hawaii conjecture. The following estimates hold:

Generalized Hawaii conjecture. If $\varkappa>\dfrac{n-1}{n}$, then

Hawaii conjecture. The number of real zeros of $Q_1[p]$ does not exceed the number of non-real zeros of $p$.

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…