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 satisfying the three-term recurrence relation of length kwith the standard initial conditions P 0 (z) = 1, P −1 (z) = • • • = P −k+1 (z) = 0 which is not a zero of A(z) lies on the real (semi)-algebraic curve C ⊂ C given by