Generalizing Tran's Conjecture

Abstract

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 $\{P_n(z){\}}_{n=1}^{\infty }$ 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 $\mathcal{C}\subset \mathbb{C}$ given by $$\Im \left(\frac{B^k(z)}{A(z)}\right)=0\mathrm{and}0\le (-1{)}^k\Re \left(\frac{B^k(z)}{A(z)}\right)\le \frac{k^k}{(k-1{)}^{k-1}}.$$ In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by $$P_n(z)+B(z)P_{n-\ell }(z)+A(z)P_{n-k}(z)=0,$$ 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 $$\Im \left(\frac{B^k(z)}{A^{\ell }(z)}\right)=0.$$

7 pages, 1 figure