Euler's Reflection Formula, Infinite Product Formulas, and the Correspondence Principle of Quantum Mechanics

TL;DR

This paper generalizes classical formulas involving gamma functions and pi, deriving new infinite product formulas and Euler's reflection formula using quantum mechanics principles without relying on traditional limit definitions.

Contribution

It introduces novel infinite product formulas for gamma function combinations and Euler's reflection formula derived via the correspondence principle, bypassing limit-based definitions.

Findings

Derived infinite product formulas including irrational numbers and nested radicals.

Established Euler's reflection formula for reciprocals of positive even integers.

Connected quantum mechanics principles with classical gamma function identities.

Abstract

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.