A Strong Notion of Universal Taylor Series

For a holomorphic function f in the open unit disc D , the N th partial sum of its Taylor series with center ζ ∈ D is denoted by S N ( f ,ζ)( z )=∑ n = 0 N ( f ( n ) ( ζ ) / n ! ) ( z − ζ ) n . Generically, all functions f in H ( D ) satisfy the following. For every compact set K ⊂ C with K ∩ D =Ø and K c connected and every polynomial h , there exists a sequence of positive integers{ λ n } ∞such that, for every 1 ∈ {0,1,2,…},sup z ∈ K|∂ l∂ z lS λ n ( f , 0 ) ( z ) − h ( l )( z )| → 0   as   n → + ∞ .