发表机构
The Ohio State University; University of Connecticut（俄亥俄州立大学; 康涅狄格大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过复兴超越级数方法研究模拟 theta 函数，证明 Mordell--Appell 积分决定其模与复兴结构，给出唯一性刻画、新函数构造及两种一致的自然边界穿越方法。
AI 中文摘要
我们发展了一种复兴超越级数方法研究模拟 theta 函数，并证明了相关的 Mordell--Appell 积分是其模结构与复兴结构的基础。通过这些积分，我们恢复了模变换律，获得了模拟 theta 向量的唯一性刻画，提供了其自然边界的规范穿越方式，并构造出新的模拟 theta 函数。在将 Mordell--Appell 积分表示为初等复兴函数的拉普拉斯变换之后，我们将拉普拉斯围道旋转至一条 Stokes 线。它们的 Stokes 现象及其模变换律重现了相应一元级数的那些性质，并结合我们的唯一性结果，重现了模拟 theta 函数的相应性质。反过来，我们将这些模变换律视为支配模拟 theta 向量的函数方程。对于 Ramanujan 的 3 阶对 (f,ω) 以及满足完整 SL(2,Z) 变换律的 5 阶、7 阶和 11 阶杰出模拟 theta 向量，我们证明了在规范最小增长归一化下，这些函数方程在单位圆盘内唯一确定一个全纯解。这个唯一解正是经典的模拟 theta 向量。我们证明了该归一化是本质性的：对于 5 阶、7 阶和 11 阶，我们确定了齐次问题的显式非平凡解，从而产生具有更快增长系数的新的模拟 theta 函数。最后，我们给出了两种内在且自然的构造来穿越自然边界 |q|=1。第一种延续模函数方程，并在另一侧取其唯一全纯解。第二种作用于尖点处的复兴渐近级数：我们将 τ 替换为 −τ，并应用 Écalle--Borel 求和。我们证明了这两种构造一致，从而给出一个规范且显式的边界穿越。
英文摘要
We develop a resurgent transseries approach to mock theta functions and show that the associated Mordell--Appell integrals underlie their modular and resurgent structure. From these integrals, we recover the modular transformation laws, obtain uniqueness characterizations of the mock theta vectors, provide a canonical crossing of their natural boundary, and produce new mock theta functions. After representing the Mordell--Appell integrals as Laplace transforms of elementary resurgent functions, we rotate the Laplace contour to a Stokes line. Their Stokes phenomena and their modular transformation laws reproduce those of the associated unary series and, combined with our uniqueness results, those of the mock theta functions. Conversely, we regard these modular transformation laws as functional equations governing the mock theta vectors. For Ramanujan's order-$3$ pair $(f,ω)$ and the distinguished mock theta vectors of orders $5$, $7$, and $11$, satisfying the full $SL(2,\mathbb Z)$ transformation laws, we prove that these functional equations uniquely determine a holomorphic solution in the unit disk under canonical minimal-growth normalization. This unique solution is precisely the classical mock theta vector. We show that the normalization is essential: for orders $5$, $7$, and $11$ we determine explicit nontrivial solutions of the homogeneous problem, yielding new mock theta functions with faster-growing coefficients. Finally, we give two intrinsic and natural constructions for crossing the natural boundary $|q|=1$. The first continues the modular functional equations and takes their unique holomorphic solutions on the other side. The second acts on the resurgent asymptotic series at the cusp: we replace $τ$ by $-τ$ and apply Écalle--Borel summation. We prove that the constructions agree, giving a canonical and explicit boundary crossing.
Comments57 pages, no figures
