A general formula for Hecke-type false theta functions

2022/12/26 by Mortenson, Eric T.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2212.13236

Abstract

In recent work where Matsusaka generalizes the relationship between Habiro-type series and false theta functions after Hikami, five families of Hecke-type double-sums of the form ( ∑_{r,s≥ 0} -∑_r,slt;0)(-1)^r+sx^ry^sq^a\binomr2+brs+c\binoms2, where b²-ac<0, are decomposed into sums of products of theta functions and false theta functions. Here we obtain a general formula for such double-sums in terms of theta functions and false theta functions, which subsumes the decompositions of Matsusaka. Our general formula is similar in structure to the case b²-ac>0, where Mortenson and Zwegers obtain a decomposition in terms of Appell functions and theta functions.

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