133 problems

Associated Hermite matching-generating-function conjecture. The integral

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…

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…

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…

Let $r\geq 3$ and let $j_0\in\{-r+1,\ldots,-2\}$. For each such initial condition, let $\{P_{j_0,n}(c)\}$ be the corresponding polynomial family. Type C ODE conjecture. For each…

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…

Hook-partition formula conjecture. Formulae for $C^{(l'+1,1)}(\alpha)$, $C^{(l+1,1)^T}(\alpha)$, $P^{(l”+1,1)}_{l”+4+k}(\alpha)$, and $P^{(l+1,1)^T}_{l+4+k}(\alpha)$ hold for all…

Chebyshev formula conjecture. Formulae for $C^{(l'+2)}(\alpha)$, $C^{(1^{l+2})}(\alpha)$, $P^{(l'+2)}_{l'+4+k}(\alpha)$, and $P^{(1^{l+2})}_{l+4+k}(\alpha)$ hold for all $l\geq -1$…

Let \boldsymbol{\textstyle\gamma}=\\{\gamma_n\}_{n\in\mathbb{Z}_+} be the sequence of Verblunsky coefficients associated with the measure $\sigma$, and let $M(\xi,\sigma)$ denote…

Let $\kappa,\tau\in\mathbb{R}$, and consider the dP$_{\rm I}^{(2)}$ equation for a sequence $\{\beta_n(\kappa,\tau)\}_{n\geq 0}$, with initial values specified by the source. A sol…

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 $\mu\in C[-1,1]$ be a nonnegative majorant, and let $\omega_\mu\in\mathcal{P}_n$ be the snake polynomial associated with $\mu$, meaning that $|\omega_\mu(x)|\leq\mu(x)$ on…

Let $K$ be a $C^{1+}$ Jordan arc, let $z0 in \overline{\mathbb{C}}backslash K$, and let $\rho:Kto[0,\infty)$ be upper-semicontinuous. Writing $tn(\rho,z0)$ for the weighted residual-pol…