Zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients

Under very general conditions on the complex coefficients of a three-term recurrence relation, it is proved that 'almost all' zeros of the polynomials generated by these relations 'accumulate' on a certain segment in the complex plane. From this result follow the convergence of diagonal Padé approximants and a generalization of Van Vleck's theorem on the convergence of $S$-fractions. Another interesting application is an extension of the so-called Nevai–Blumenthal class of polynomials $M(a,2b)$ to the case when $a,b\in{\mathbb C}$.