Vector-valued modular resurgent series are defined, and Eichler integrals of unary theta series are proven to be non-trivial examples whose Stokes constants are rotated by the modular S-matrix and whose median resummation reconstructs the series.
A Framework for Modular Properties of False Theta Functions
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abstract
False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the Circle Method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions proposed in this paper.
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Modular resurgent structures for vectors
Vector-valued modular resurgent series are defined, and Eichler integrals of unary theta series are proven to be non-trivial examples whose Stokes constants are rotated by the modular S-matrix and whose median resummation reconstructs the series.