A curious identity arising from Stirling's formula and saddle-point method on two different contours

Hsien-Kuei Hwang¹

¹Academia Sinica

The electronic journal of combinatorics, Tome 30 (2023) no. 4

Résumé

We prove the curious identity in the sense of formal power series:\[\int_{-\infty}^{\infty}[y^m]\exp\left(-\frac{t^2}2+\sum_{j\ge3}\frac{(it)^j}{j!}\, y^{j-2}\right)\mathrm{d} t= \int_{-\infty}^{\infty}[y^m]\exp\left(-\frac{t^2}2+\sum_{j\ge3}\frac{(it)^j}{j}\, y^{j-2}\right)\mathrm{d} t,\]for $m=0,1,\dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$, which arises from applying the saddle-point method to derive Stirling's formula. The generality of the same approach (saddle-point method over two different contours) is also examined, together with some applications to asymptotic enumeration.

Détail

DOI : 10.37236/11785

Classification : 05A16, 05A15, 41A60, 33B15
Mots-clés : saddle-point method, Cauchy's integral representation