AI 中文总结
本文对拉马努金整函数 $A_q(z)$ 在特定转折点标度下进行渐近分析，得到其艾里函数展开式及一阶修正，确定了其正零点与艾里零点的对应关系，验证了相关差分方程与微分方程的联系，补充了森田反对称伴随函数的归一化极限结果。
AI 中文摘要
拉马努金整函数定义为 $A_q(z)=\sum_{k=0}^{\infty}\frac{q^{k^2}}{(q;q)_k}(-z)^k$，本文在转折点标度 $q=e^{-\varepsilon}$、$z=\frac{\sqrt q}{4}e^{-\varepsilon^{2/3}\zeta}$（其中 $\zeta$ 属于复平面的紧子集）下研究该函数。经显式指数归一化后，通过直接合并鞍点分析，在紧子集上一致得到展开式 $\operatorname{Ai}(\zeta)+\frac{\varepsilon^{2/3}}{30}\left(4\zeta\operatorname{Ai}(\zeta)+\zeta^2\operatorname{Ai}'(\zeta)\right)+O_K(\varepsilon)$，其中一阶修正项显式给出并带有定量余项。对每个固定的 $n$，相同分析可确定与第 $n$ 个艾里零点相关的正零点，并证明其全局为 $A_q$ 的第 $n$ 个正零点。对归一化 $q$-差分方程展开可重现艾里微分方程并验证标度的正确性。附录记录了森田（Morita）的反对称伴随函数，其归一化极限为 $\operatorname{Bi}$，以及该函数的单值亚纯下降，数值表格展示了归一化、修正项和零点公式。
英文摘要
Let $$ A_q(z)=\sum_{k=0}^{\infty}\frac{q^{k^2}}{(q;q)_k}(-z)^k $$ be Ramanujan's entire function. We study it in the turning-point scaling $$ q=e^{-\varepsilon},\qquad z=\frac{\sqrt q}{4}e^{-\varepsilon^{2/3}ζ}, $$ with $ζ$ in a compact subset of $\mathbb{C}$. After an explicit exponential normalization, a direct coalescing-saddle analysis gives, uniformly on compact subsets, the expansion $$ \operatorname{Ai}(ζ) +\frac{\varepsilon^{2/3}}{30} \left(4ζ\operatorname{Ai}(ζ) +ζ^2\operatorname{Ai}'(ζ)\right) +O_K(\varepsilon). $$ Thus the first correction is explicit and comes with a quantitative remainder. For every fixed $n$, the same analysis locates the positive zero associated with the $n$-th Airy zero and proves that it is globally the $n$-th positive zero of $A_q$. Expanding the normalized $q$-difference equation recovers the Airy differential equation and confirms the scaling. An appendix records Morita's antisymmetric companion, whose normalized limit is $\operatorname{Bi}$, together with a single-valued meromorphic descent of it. Numerical tables illustrate the normalization, the correction, and the zero formulas.
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