UNINHIBITED POINCARÉ SERIES

Abstract

We introduce a class of functions that generalize the epoch-making series of Poincaré and Petersson. Our "uninhibited Poincaré series" permits both a complex weight and an arbitrary multiplier system that is independent of the weight. In this initial paper we provide their Fourier expansions, as well as their modular behavior. We show that they are modular integrals that possess interesting periods. Moreover, we establish with relative ease that they "almost never" vanish identically. Along the way we present a seemingly unknown historical truth concerning Kloosterman sums, and also an alternative approach to Petersson's factor systems. The latter depends upon a simple multiplication rule.