Airy Turning-Point Asymptotics for Ramanujan's Entire Function $A_q(z)$

Abstract

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 $$q=e^{-\epsilon },z=\frac{\sqrt{q}}{4}{e}^{-{\epsilon }^{2/3}\hat{I}\P },$$ with $\hat{I}\P$ in a compact subset of $\mathbb{C}$. After an explicit exponential normalization, a direct coalescing-saddle analysis gives, uniformly on compact subsets, the expansion $$\mathrm{Ai}(\hat{I}\P )+\frac{{\epsilon }^{2/3}}{30}\left(4\hat{I}\P \mathrm{Ai}(\hat{I}\P )+\hat{I}{\P }^2{\mathrm{Ai}}^{\prime }(\hat{I}\P )\right)+O_K(\epsilon ).$$ 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 $\mathrm{Bi}$, together with a single-valued meromorphic descent of it. Numerical tables illustrate the normalization, the correction, and the zero formulas.

31 pages

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