On the location of ratios of zeros of special trinomials

Given coprime integers $k,\mathrm{\ell }$ with $k>\mathrm{\ell }⩾1$ and arbitrary complex polynomials $A(z),B(z)$ with $\mathrm{°}(A(z)B(z))⩾1$, we consider the polynomial sequence $\{P_n(z)\}$ satisfying a three-term recurrence $P_n(z)+B(z)P_{n-\mathrm{\ell }}(z)+A(z)P_{n-k}(z)=0$ subject to the initial conditions $P_0(z)=1$, $P_{-1}(z)=\cdots =P_{1-k}(z)=0$ and fully characterize the real algebraic curve $\mathrm{\Gamma }$ on which the zeros of the polynomials in $\{P_n(z)\}$ lie. In addition, we show that, for any (randomly chosen) $n\in {\mathbb{Z}}_{⩾1}$ and zero $z_0$ of $P_n(z)$ with $A(z_0)\mathrm{\ne }0$, at-least two of the distinct zeros of the trinomial $D(t;z_0):=A(z_0)t^k+B(z_0)t^{\mathrm{\ell }}+1$ have a ratio that lies on the real line and / or on the unit circle centred at the origin. This reveals a previously unknown geometric property exhibited by the zeros of trinomials of the form $t^k+a{t}^{\mathrm{\ell }}+1$ where $a\in \mathbb{C}-\{0\}$ is such that $a^k\in \mathbb{R}$.