A curious identity in connection with saddle-point method and Stirling's formula

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 $f$. The generality of this identity from the perspective of saddle-point method is also examined.