A Further Generalisation of Sums of Higher Derivatives of the Riemann Zeta Function

We prove an asymptotic for the sum of ^ (n) () X^ where ^ (n) (s) denotes the nth derivative of the Riemann zeta function, X is a positive real and denotes a non-trivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height T, as T. We also specify what the asymptotic formula becomes when X is a positive integer, highlighting the differences in the asymptotic expansions as X changes its arithmetic nature.