Hermite multiplier sequences and their associated operators

We provide an explicit formula for the coefficient polynomials of a Hermite diagonal differential operator. The analysis of the zeros of these coefficient polynomials yields the characterization of generalized Hermite multiplier sequences which arise as Taylor coefficients of real entire functions with finitely many zeros. We extend our result to functions in $\mathcal{L-P}$ with infinitely many zeros, under additional hypotheses.