  We generalize the derivation of the Wallis formula for $\pi$ 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.
