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

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 $\pi /4,2,\sqrt{2+\sqrt{2}},2\pi /3\sqrt{3},$ 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 $e^{2/3}/\sqrt{3}$ of Pippenger's 1980 infinite product for $e/2$, and we conjecture that they can be generalized to a product for a power of $e^{1/K}$.

7 pages, accepted by Amer. Math. Monthly