4 problems

Optimal-measure claim. For any fixed $n\geq 1$, the optimal measure is $d\sigma=w^\lambda dx$ for some $\lambda\in(-1/2,\infty)$. The value of $\lambda$ may depend on $n$.

Let $G$ be a graphical function for which the radial representation and Gegenbauer expansion are defined. Universal Gegenbauer expansion conjecture. The Gegenbauer expansion (gfr)…

Positivity conjecture. $F_n^{\lambda,\delta}(t)>0$ for every $t\in(0,\pi]$ if $\delta\geq\lambda+1$.

Integral positivity conjecture. The inequality $F_n^{\lambda,\delta}(t)>0$ holds for every $n\in\mathbb{N}_0$ and every $t\in(0,\pi]$ if and only if $\delta\geq\lambda+1$.