7 problems

Monotonicity conjecture. The sequence $\{K_i\}_{i=1}^{\infty}$ is monotonically increasing and positive, and satisfies

Let $\bar z_i$ be the secondary intermediate nodes and let the first-order partial moments be defined by $\alpha_0=0$ and $\alpha_i=\alpha_{i-1}-2x_iw_i$. Partial-moment conjecture…

Monotonicity conjecture. The sequence $\{E_i\}_{i=1}^{\infty}$ is negative, decreasing, and satisfies

Let $\{x_i\}$ be the primary Gauss–Legendre nodes and $\{\bar z_i\}$ the secondary intermediate nodes. Asymptotic interlacing conjecture. The primary nodes and secondary intermedia…

Let $x_i$ and $z_i$ be respectively the primary and secondary nodes of the $n$-point Gauss–Legendre quadrature, and let $\kappa$ be the asymptotic parameter. Uniform primary–second…

Let $j_{k}$ denote the positive zeros of the Bessel function $J_1$, and define, for $i\geq 0$, … where the sum is zero for $i=0$. Monotonicity conjecture. The sequence…

Let $x_i$ and $w_i$ be respectively the nodes and weights of the $n$-point Gauss–Legendre quadrature, and let $\kappa$ be the parameter used in the quadrature asymptotics. Define ……