The root distribution of polynomials with a three-term recurrence

For any fixed positive integer n  , we study the root distribution of a sequence of polynomials Hm(z)Hm(z) satisfying the rational generating function∑m=0∞Hm(z)tm=11+B(z)t+A(z)tn where A(z)A(z) and B(z)B(z) are any polynomials in z   with complex coefficients. We show that the roots of Hm(z)Hm(z) which satisfy A(z)≠0A(z)≠0 lie on a specific fixed real algebraic curve for all large m.