We generalize the derivation of the Wallis formula for π from a variational computation of the spectrum of the hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational numbers such as $\sqrt{2}$ as well as some nested radicals. We also derive Euler’s reflection formula for reciprocals of positive even integers. We show that Bohr’s correspondence principle allows us to derive our product formulas and the reflection formula without the need for the limit definition of the gamma function.

中文翻译：

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欧拉反射公式、无穷积公式、量子力学对应原理

我们从氢原子光谱的变分计算中推广了π的沃利斯公式的推导。我们为伽马函数的某些组合获得无限乘积公式，其中包括无理数，例如$\sqrt{2}$以及一些嵌套的部首。我们还推导出正偶数的倒数的欧拉反射公式。我们表明，玻尔对应原理允许我们推导出乘积公式和反射公式，而无需伽马函数的极限定义。