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⇱ Ramanujan's Hypergeometric Identity -- from Wolfram MathWorld

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Ramanujan's Hypergeometric Identity

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👁 sum_(n=0)^(infty)(-1)^n[((2n-1)!!)/((2n)!!)]^3	👁 =	👁 1-(1/2)^3+((1·3)/(2·4))^3+...	(1)
👁 Image	👁 =	👁 _3F_2(1/2,1/2,1/2; 1,1;-1)	(2)
👁 Image	👁 =	👁 [_2F_1(1/4,1/4; 1;-1)]^2	(3)
👁 Image	👁 =	👁 (Gamma^2(9/8))/(Gamma^2(5/4)Gamma^2(7/8)),	(4)

where 👁 _2F_1(a,b;c;x)
is a hypergeometric function, 👁 _3F_2(a,b,c;d,e;x)
is a generalized hypergeometric function, and 👁 Gamma(z)
is a gamma function.

See also

Generalized Hypergeometric Function, Hypergeometric Function

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References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 106, 1999.

Referenced on Wolfram|Alpha

Ramanujan's Hypergeometric Identity

Cite this as:

Weisstein, Eric W. "Ramanujan's Hypergeometric Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RamanujansHypergeometricIdentity.html

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