13 problems

Let $P_n^{(\alpha,\beta)}(x)$ denote the Jacobi polynomial of degree $n$, with $n\in\mathbb{N}$, $\alpha>-1$, and $\beta>-1$. Askey's conjecture. The zeros of…

For $k\geq0$ and $\alpha,\beta\geq-\frac12$, let … In the parameter range $\alpha\geq\beta$, the source had obtained an upper bound of order $O(\alpha^{2/3}(1+\alpha/k)^{1/3})$. Kr…

Jacobi zero-preservation conjecture. If $\alpha$ and $\beta$ are non-negative integers, then $T_{\alpha,\beta}$ preserves the set of polynomials whose zeros lie only in $(-1,1)$. T…

Orthogonality conjecture. Solutions $f\in\mathcal{A}$ of the Hardy-type integral equation (5.2) are orthogonal with respect to their own zeros for the Jacobi weight.

Erdős–Magnus–Nevai conjecture. The real order of $\mathcal{M}_k^{\alpha,\beta}$ is $O(\alpha^{1/2})$. This was proposed after an upper bound of order…

Erdélyi–Magnus–Nevai conjecture. The quantity $\mathcal{M}_k^{\alpha,\beta}$ satisfies

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…

Let $C$ and $D$ be two Type II codes of length $n$ over $\F_{2}$, and let $\bm{w}\in\F_{2}^{n}$. A code is $t$-homogeneous when, for every nonzero weight $w$, the supports of its c…

Simple-zero conjecture. For any partitions $\lambda$ and $\mu$, take $\alpha>-1$ and $\beta>m_1$ such that the stated conditions are satisfied. If $\lambda$ is an even partition, t…

Let $\alpha,\beta\in(0,\infty)\setminus\mathbb{N}$, and let $F_n^{(\alpha,\beta),\delta}(t)$ denote the Jacobi-polynomial integral defined earlier in the paper. For $t\in(0,\pi]$ a…

Let $P_n^{(\alpha,\beta)}$ denote the Jacobi polynomial of degree $n$, and let $n\in\mathbb{N}$, $\alpha>-1$, and $-2<\beta<-1$. The zeros of two polynomials are called interlacing…

Let $m\in\mathbb{N}$, and consider the theorem asserting that all critical points of the polynomial $Q_n$ are simple, lie in $[-1,1]$, and interlace the zeros of $L_n$ for the Jaco…

Askey–Gasper conjecture. For $r>-1$ and $-1<x\leq 1$,