Zero Distributions of Derivatives of Polynomial Families Centering on a Set

Suppose C ⊂ C C ⊂ C is compact. Let qₖ q k be a sequence of polynomials of degree nₖ → ∞ n k → ∞ , such that the locus of roots of all the polynomials is bounded, and the number of roots of qₖ q k in any closed set L disjoint from C is uniformly bounded. Supposing that (qₖ)ₖ ( q k ) k has an asymptotic root distribution μ μ we provide conditions on C and μ μ assuring the sequence of m th derivatives (qₖ⁽ᵐ⁾)ₖ ( q k ( m ) ) k has the same asymptotic root distribution μ μ for any m≥ 1 m ≥ 1 . This complements recent results of Totik, [19].