Around a Conjecture of K. Tran

Abstract: We study the root distribution of a sequence of polynomials ${P_n(z)}{n=0}^{\infty}$ with the rational generating function $$ \sum{n=0}^{\infty} P_n(z)t^n= \frac{1}{1+ B(z)t^\ell +A(z)t^k}$$ for $(k,\ell)=(3,2)$ and $(4,3)$ where $A(z)$ and $B(z)$ are arbitrary polynomials in $z$ with complex coefficients. We show that the zeros of $P_n(z)$ which satisfy $A(z)B(z)\neq 0$ lie on a real algebraic curve which we describe explicitly.