On Uniqueness of Mock Theta Functions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918463972311040 |
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| author | Costin, Ovidiu Dunne, Gerald V. Saraeb, Ali |
| author_facet | Costin, Ovidiu Dunne, Gerald V. Saraeb, Ali |
| contents | 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 $π$, 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 $\hat q=e^{-πi τ}$ and $\hat q_1=e^{-πi (-1/τ)}$.
We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{πi τ}$ and $q_1=e^{πi (-1/τ)}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20034 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Uniqueness of Mock Theta Functions Costin, Ovidiu Dunne, Gerald V. Saraeb, Ali Number Theory High Energy Physics - Theory Mathematical Physics Classical Analysis and ODEs 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 $π$, 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 $\hat q=e^{-πi τ}$ and $\hat q_1=e^{-πi (-1/τ)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{πi τ}$ and $q_1=e^{πi (-1/τ)}$, 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. |
| title | On Uniqueness of Mock Theta Functions |
| topic | Number Theory High Energy Physics - Theory Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.20034 |