A Curious Identity Arising From Stirling's Formula and Saddle-Point Method on Two Different Contours

Abstract

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.