7 problems

Numerical-radius conjecture. If

Legendre-function conjecture. These Legendre polynomials can be used as GKN conditions to define the $n/2$ left-definite domain, and together the $2n$ functions constitute a basis…

Increasing-density conjecture. For any given $t>0$, $f_d(t)$ is an increasing function of $d$.

Let $f(x)=\sum_{k=0}^n a_kx^k\in\mathbb{R}[x]$ have all its zeros in $(-1,1)$, and define the linear transformation $T$ by … where $P_k(x)$ is the $k$-th Legendre polynomial. Fisk'…

Legendre expansion error-rate conjecture. The error has the following properties: (1) for $x\in(-1,a)\cup(a,1)$, $\varepsilon(x)\leq C(x)p^{-\alpha}$ with $\alpha=-(\beta+1)$, wher…

Coefficient-formula conjecture. The coefficients satisfy

Let $P_n(x)$ denote the Legendre polynomial of degree $n$. Stieltjes's irreducibility conjecture. For every $n$, the polynomials … are irreducible. This is a stronger conjecture th…