New Wallis- and Catalan-Type Infinite Products for $\pi, e$, and $\sqrt{2+\sqrt2}$

TL;DR

This paper introduces new infinite products of Wallis- and Catalan-type for fundamental constants like π, e, and nested radicals, extending classical formulas using gamma and Stirling's functions.

Contribution

It generalizes classical infinite products for π and e, providing new formulas and conjectures for powers of e, using gamma and Stirling's functions.

Findings

New Wallis-type products for π/2, π/4, 2, and nested radicals.

Catalan-type products for e/4, √e, and e^{3/2}/2.

An analog for e^{2/3}/√3 and conjectures for further generalizations.

Abstract

We generalize Wallis's 1655 infinite product for $\pi/2$ to one for $(\pi/K)\csc(\pi/K)$, as well as give new Wallis-type products for $\pi/4, 2, \sqrt{2+\sqrt2}, 2\pi/3\sqrt3,$ and other constants. The proofs use a classical infinite product formula involving the gamma function. We also extend Catalan's 1873 infinite product of radicals for $e$ to Catalan-type products for $e/4,\sqrt{e}$, and $e^{3/2}/2$. Here the proofs use Stirling's formula. Finally, we find an analog for $e^{2/3}/\sqrt3$ of Pippenger's 1980 infinite product for $e/2$, and we conjecture that they can be generalized to a product for a power of $e^{1/K}$.