On hyperderivatives of single-cuspidal Drinfeld modular forms with A-expansions

We show that the Drinfeld modular forms with A-expansions that have been constructed by the author are precisely the hyperderivatives of the subfamily of single-cuspidal Drinfeld modular forms with A-expansions that remain modular after hyperdifferentiation. In addition, we show that certain Drinfeld-Poincaré series display a similar behavior with respect to hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms with A-expansions are Drinfeld-Poincaré series. The Drinfeld-Poincaré series that we consider generalize previous examples of such series by Gekeler, and Gerritzen and van der Put.