135 problems

Associated Hermite matching-generating-function conjecture. The integral

For every degree $n$, the critical vectors occurring in the unconditional part of Chenevier's automorphic Hermite--Minkowski theorem are strictly positive in the associated orthogo…

Let $H_n$ be the degree-$n$ Hermite polynomial, and let $0<x_{n,1}<\cdots<x_{n,\lfloor n/2\rfloor}$ denote its positive zeros. Prove that for every integer $n\ge 4$,…

Positive-integrality conjecture. For $\beta\in\{1,2\}$, the expansion coefficients satisfy

Ismail's characterization conjecture. Then $(P_n)_{n\geq 0}$ are continuous $q$-Jacobi polynomials, Al-Salam–Chihara polynomials, or special or limiting cases of these families. Mo…

Ignjatović's conjecture. If

Root-distribution conjecture. For each $k\geq -1$, the polynomial $P^{(\lambda)}_{l+4+k}$ is square free. Furthermore, for sufficiently large $k$, it has a fixed number of roots ou…

Let $d$ be a positive integer and $a\geq0$, and consider the hermitian inner product with density … For each nonnegative integer $N$, let $G_N=((z^n,z^m))_{n,m=0}^N$ be its Gram ma…

Let $A,A^*$ be a tridiagonal pair on a finite-dimensional vector space $V$ over an algebraically closed field $\mathbb K$. Let $d$ be its diameter, and let…

Let $w$ be the density of the absolutely continuous part of a measure on the unit circle, let $\alpha=(\alpha_n)_{n\geq 0}$ be its Verblunsky-coefficient sequence, and let $S$ be t…

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…

Let $C^{(l+1,1)}(\alpha)$, $C^{(l+1,1)^T}(\alpha)$, $P_{l+4+k}^{(l+1,1)}$, and $P_{l+4+k}^{(l+1,1)^T}$ denote the polynomials and Chebyshev series defined in the preceding formulae…

DMPK limiting-density conjecture. The density $\rho(x;\mathbb{M})$ is related to $\psi(x)$ by

Let $\{P_n\}_{n=0}^\infty$ be an orthogonal polynomial system with respect to an arbitrary positive measure $d\mu=W\,dx$ supported on an interval $I$ of the real line. Assume that…

Matrix Stahl–Totik conjecture. Then $\mu$ is regular.

Krasikov's stronger conjecture. One has

Let $\{p_n(z)\}_{n\in\mathbb N}$ be defined by … where $p_{-2}(z)=p_{-1}(z)=0$, $p_0(z)=1$, and $C\in\mathbb R$. Egecioglu–Redmond–Ryavec 3-conjecture. All polynomials in the seque…

Let $V$ be a finite-dimensional vector space over $\mathbb K$ of positive dimension, and let $A,A^*,A^\varepsilon:V\to V$ be linear transformations. A Leonard pair is a pair of tra…

Matrix-valued product-expansion conjecture. The product admits a unique expansion of this form, with matrix coefficients $A_k$, for all nonnegative $n$.

Log-squared weight conjecture. The weight satisfies

Let $\Theta_k$ index the multivariate Hermitian Jacobi polynomials $\tilde{P}^{(\alpha,\beta)}_{\tau_1,\dots,\tau_k;\phi}$, defined by triangular expansion in the multivariate Schu…

Let $\mathcal{G}_{\kappa_1,\dots,\kappa_{k+1}}^m$ be the Hermitian Jacobi operator on the simplex, and let the multivariate Schur polynomials be the basis of unitary-conjugation-in…

Let $m$ be a positive integer, let $k$ be the number of matrix variables, let $(\kappa_1,\dots,\kappa_{k+1})$ be the parameters of the Hermitian Jacobi operator, and let…

Eventual real-rootedness conjecture. For all sufficiently large $n$, every zero of $q_n^{u,v}$ is real.

Let $A=\mathbb{C}[x^{\pm1},u\mid u^2=P(x)]$, where $P(x)\in\mathbb{C}[x]$ is palindromic of even degree $2r$ with simple roots. Let $A/\partial A$ denote the quotient by exact deri…