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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.

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