On Uniqueness of Mock Theta Functions

TL;DR

This paper introduces a resurgent approach to uniquely continue mock theta functions across their natural boundary, establishing a canonical extension and identifying distinguished families for orders 3 and 5.

Contribution

It develops a novel resurgent method to prove the uniqueness of mock theta functions' continuation across their boundary, extending to higher orders in future work.

Findings

Proves unique continuation of mock theta functions using resurgent analysis.

Establishes mock-modular relations on the Stokes line for known cases.

Identifies a canonical family of mock theta functions for orders 3 and 5.

Abstract

We develop a resurgent approach to the problem of unique continuation of mock theta functions across their natural boundary. The starting point is the representation of the associated Mordell-Appell integrals as Laplace transforms of resurgent functions, which serve as the primary analytic objects. By rotating the Laplace contour by $\pi$, i.e. onto the Stokes line, one obtains, in all known cases, the mock-modular relations between the Mordell-Appell integrals and the corresponding unary series in $\hat q=e^{-\pi i \tau}$ and $\hat q_1=e^{-\pi i (-1/\tau)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{\pi i \tau}$ and $q_1=e^{\pi i (-1/\tau)}$, with coefficients determined by the corresponding Mordell-Appell integrals. This yields a canonical continuation across the natural boundary, given by a resurgent extension of the classical principle of permanence of relations, and singles out a distinguished family of mock theta functions in each group. We present a complete analysis for the order 3 and 5 cases (mf3 and mf5). The method extends naturally to higher orders; a general theory will appear in a separate paper.