3 problems

Zero-location conjecture. For $n<p_0$, $T^{(3)}_{n-1}(u,v)$ has two real zeros $\pm\zeta^{(3)}_{n-1}$ for some $\zeta^{(3)}_{n-1}\in\mathbf R_{>0}$, while for $n=p_0$ it has a doub…

Zero-location conjecture. All the zeros of $T^{(1)}_{n-1}(u,v)$ are located on the almost straight lines

Let $u$ be sufficiently small, with $|u|\ll 1$, let $s$ be a positive integer, and let $a\in\{1,2,\dots,s\}$. For each positive integer $m$, let $T^{(a)}_{m}(v)$ denote the transfe…