Disjoint universality connected with differential operators

For a simply connected domain G, we study the problem of disjoint universality for the sequences of operators T α , n : H ( G ) → H ( G ) , defined by T α , n ( f ) ( z ) = ∑ j = 0 n f ( j ) ( z ) j ! ( α z ) j , where α ∈ C ∖ { 0 } . Note that T α , n ( f ) ( z ) is the n t h partial sum of the Taylor expansion of f around z on ( α + 1 ) z . The motivation to study such sequences comes from universal Taylor series, by changing the role of the center of expansion.