Generalizing Tran's Conjecture

A conjecture of Khang Tran [6] claims that for an arbitrary pair of polynomials $A(z)$ and $B(z)$, every zero of every polynomial in the sequence $\{P_n(z){\}}_{n=1}^{\mathrm{\infty }}$ satisfying the three-term recurrence relation of length $k$ $$P_n(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0$$ with the standard initial conditions $P_0(z)=1$, $P_{-1}(z)=\cdots =P_{-k+1}(z)=0$ which is not a zero of $A(z)$ lies on the real (semi)-algebraic curve $\mathcal C \subset \mathbb {C}$givenby$$\Im \left(\frac{B^k(z)}{A(z)}\right)=0\quad {\rm and}\quad 0\le (-1)^k\Re \left(\frac{B^k(z)}{A(z)}\right)\le \frac{k^k}{(k-1)^{k-1}}.$$Inthisshortnote,weshowthatfortherecurrencerelation(generalizingthelatterrecurrenceofTran)givenby$$P_n(z)+B(z)P_{n-\ell}(z)+A(z)P_{n-k}(z)=0, with coprime $k$ and $\ell$ and the same standard initial conditions as above, every root of $P_n(z)$ which is not a zero of $A(z)B(z)$ belongs to the real algebraic curve $\mathcal C_{\ell,k}$ given by\Im \left(\frac{B^k(z)}{A^\ell(z)}\right)=0.$$