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

Abstract

We show that the Drinfeld modular forms with $A$-expansion 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 Drinfeld-Poincar\'{e} series display a similar behavior with respect to hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms with $A$-expansions are Drinfeld-Poincar\'{e} series. The Drinfeld-Poincar\'{e} series that we consider generalize previous examples of such series by Gekeler, and Gerritzen and van der Put.