The asymptotic expansion of a hypergeometric series coming from mirror symmetry (Q901383)

The hypergeometric series \[ \mathcal F_n(w,x)=\sum_{d=0}^\infty x^d\frac{\prod_{r=1}^{nd}(nw+r)}{\prod_{r=1}^{d}((w+r)^n-w^n)} \] appears in the theory of Calabi-Yau manifolds, and, in particular, the case \(n=5\) and \(w=0\) appears in string theory. D. Zagier and A. Zinger had shown that \(\mathcal F_n(w,x)\) has the asymptotic expansion \[ \mathcal F_n(w,x)\sim e^{\mu(x)w}\sum_{s=0}^\infty\Phi_s(x)w^{-s} \] as \(w\to\infty\). The function \(\mu\) is explicitly given, but the determination of the functions \(\Phi_s\) is cumbersome if we follow the Zagier-Zinger method. The present author gives a more convenient determination of \(\Phi_s\) from where the asymptotics of \(\Phi_s\) is more apparent, too.