Euler’s reflection formula, infinite product formulas, and the correspondence principle of quantum mechanics

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