New Wallis- and Catalan-type infinite products for \(\pi\), \(e\), and \(\sqrt{2 + \sqrt 2}\). (English) Zbl 1216.40002

Wallis’ infinite product is generalized and other Wallis-type products for other constants are established. The proofs use a classical infinite product formula involving the gamma function. Using the Stirling formula, Catalan’s infinite product of radicals for \(e\) is extended to Catalan-type products for other constants involving \(e\). Finally, an analog of Pippenger’s formula is given.


40A20 Convergence and divergence of infinite products
33B15 Gamma, beta and polygamma functions
11Y60 Evaluation of number-theoretic constants
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Online Encyclopedia of Integer Sequences:

a(n) = 2*n*3^(n-1) - (3^n-1)/2.