Scaled generalized Bernstein polynomials and containment regions for the zeros of polynomials

摘要

Let Pn(x)=∑k=0nβkbk,n(x;α,β),βn≠0, where bk,n(x;α,β)≔(α+x)k(β-x)n-k,k=0,1,…,n,α,β∈C,α≠-β, a generalized scaled Bernstein polynomial. Extending a result of Winkler [A resultant matrix for scaled Bernstein polynomials, Linear Algebra Appl. 319 (2000) 179–191] a companion matrix for Pn is given. The application of some matrix methods to this companion matrix yields regions for the zeros of Pn.