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Pippenger Product

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The Pippenger product is an unexpected Wallis-like formula for e given by

e/2=(2/1)^(1/2)(2/34/3)^(1/4)(4/56/56/78/7)^(1/8)...
(1)

(OEIS A084148 and A084149; Pippenger 1980). Writing the product as product_(n=1)^(infty)a_n^(1/2^n), the rational bases satisfy a_1=2 and, for n>=2,

a_n=([(2^(n-1)-1)!!]^2[(2^n)!!]^2)/(2[(2^(n-1))!!]^2[(2^n-1)!!]^2)
(2)
=(2^(2^n-1)[Gamma(1/2+2^(n-2))]^4)/(pi[Gamma(1/2+2^(n-1))]^2),
(3)

where z!! is a double factorial and Gamma(z) is the gamma function.

See also

e, Wallis Formula

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References

Pippenger, N. "An Infinite Product for e." Amer. Math. Monthly 87, 391, 1980. https://doi.org/10.1080/00029890.1980.11995044.Sloane, N. J. A. Sequences A084148 and A084149 in "The On-Line Encyclopedia of Integer Sequences."

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Pippenger Product

Cite this as:

Weisstein, Eric W. "Pippenger Product." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PippengerProduct.html

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