New Wallis- and Catalan-Type Infinite Products for α, e and

Abstract

We generalize Wallis’s 1655 infinite product for α/2 to one for (α/K)csc(α/K), as well as give new Wallis-type products for α/4, , , 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, and e3/2/2. Here the proofs use Stirling’s formula. Finally, we find an analog for , of Pippenger’s 1980 product for e/2, and conjecture that they can be generalized to a product for a power of e1/K.