2026/07/29 by Yu-Tian Li
Mathematics · #math.CA #math.CV #msc:33D15 #msc:33C10 #msc:39A13 #msc:41A60
31 pages
arxiv created 2026/07/29 · arxiv updated 2026/07/31
Let Aq(z)=∑k=0∞\fracqk2(q;q)k(-z)k be Ramanujan's entire function. We study it in the turning-point scaling q=e-ε, z=(√ q)/(4)e^-ε2/3ζ, with ζ in a compact subset of ℂ. After an explicit exponential normalization, a direct coalescing-saddle analysis gives, uniformly on compact subsets, the expansion Ai(ζ) +\fracε2/330 (4ζAi(ζ) +ζ2Ai'(ζ)) +OK(ε). 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 Aq. 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 Bi, together with a single-valued meromorphic descent of it. Numerical tables illustrate the normalization, the correction, and the zero formulas.