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Ramanujan's Hypergeometric Identity


sum_(n=0)^(infty)(-1)^n[((2n-1)!!)/((2n)!!)]^3=1-(1/2)^3+((1·3)/(2·4))^3+...
(1)
=_3F_2(1/2,1/2,1/2; 1,1;-1)
(2)
=[_2F_1(1/4,1/4; 1;-1)]^2
(3)
=(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 Web Resource. https://mathworld.wolfram.com/RamanujansHypergeometricIdentity.html

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