论文标题
与鞍点法和斯特林公式有关的一个奇怪身份
A curious identity in connection with saddle-point method and Stirling's formula
论文作者
论文摘要
我们证明了正式力量系列的好奇身份：\ [\ int _ { - \ infty}^{\ infty} [y^m] \ exp \ left（ - \ frac {t^2} 2 +sum_ sum_ {j \ ge3}} y^{j-2} \ right）\ mathrm {d} t = \ int _ { - \ infty}^{\ infty} [y^m] \ exp \ left（ - \ frac {t^2} 2+ \ sum__ y^{j-2} \ right）\ mathrm {d} t，\]对于$ m = 0,1，\ dots $，其中$ [y^m] f（y）$表示$ f $的泰勒扩展中的$ y^m $的系数。还研究了从鞍点方法的角度来看，这种身份的一般性。
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.
