13 problems

Tran's conjecture. All zeros of every polynomial in the sequence lie on $\Gamma$, and these roots become dense in $\Gamma$ as $n\to\infty$.

Let $\{p_n(x)\}_{n=m}^{\infty}$ be a family of exceptional orthogonal polynomials with weight function … where $W_0(x)$ is the weight function of a classical orthogonal polynomial…

Let $p_n$ be the polynomials defined by formula (20) or, equivalently, by the recurrent relation (10) in the source. Reality conjecture. All zeros of $p_n$ are real. The distributi…

Monotonicity conjecture. Taking into account the stated interlacing relations, $q_{n,k}^{(Hj)}$ is an increasing function of $mbda$ for each $k$.

For each positive integer $m$, let … and let $z_k$, $1\leq k\leq m$, be the zeros of $P_m(z)$. Boros' zero-distribution conjecture. The zeros satisfy … This conjecture concerns the…

For $n=0,1,\dots$ and $\alpha>-1$, define the transitional Laguerre polynomials $\widehat{L}^{(\alpha)}_{n+1}(\cdot,t)$ by … where … Let the extreme zeros denote the smallest and l…

For $n=0,1,\dots$ and $\alpha>-1$, define the transitional Laguerre polynomials $\widehat{L}^{(\alpha)}_{n+1}(\cdot,t)$ by … where … Let the extreme zeros mean the smallest and lar…

Let $P(z)=\sum_{n=0}^N a_nz^n$ be a Littlewood polynomial of degree $N$, with $a_n\in\{\pm1\}$, and call it reciprocal if $P(z)=z^NP(1/z)$. Let $Z_{\mathcal{L}}(N)$ be the minimum…

Let $\lambda=(\lambda_1,\dots,\lambda_m)\in\mathbb{N}^m$ be a partition with $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_m\geq1$, and define the generalized Hermite polynomial … w…

Orthogonality-grabbing conjecture. This phenomenon occurs because the spanning functions $\{p_k(z)\}$ are orthogonal on the unit disk, and this orthogonality is sufficiently strong…

Let $f$ be an entire function of positive, finite order $\lambda$, and let $p_n(z)$ denote its $n^\text{th}$ partial sum. Edrei–Saff–Varga's modified width conjecture. There exist…

Saff–Varga's width conjecture. There exists an infinite sequence of positive integers $N$ such that no region $S_\theta(\tau)$ is devoid of all zeros of all partial sums $p_n(z)$ w…

Let the coefficients of the moments $m_{2l}$ be regarded as polynomials in the ensemble parameter $\kappa$, after extracting their stated powers of $N$ and $\kappa$. The coefficien…