Ramanujan's Hypergeometric Identity

Author

Eric W. Weisstein
May 19, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/RamanujansHypergeometricIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RamanujansHypergeometricIdentity.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Check

V5

Sum[(-1)^k*(Pochhammer[a,k]^3/Pochhammer[1,k]^3)*z^k,
{k,0,Infinity}]

Sum[(-1)^n((((((2n-1)!!)/((2n)!!))))^3),{n,0,∞}]

V7

Sum[(-1)^k*(Pochhammer[a,k]^3/Pochhammer[1,k]^3)*z^k,
{k,0,Infinity}]

Sum[(-1)^n((((((2n-1)!!)/((2n)!!))))^3),{n,0,∞}]

<<Utilities`Simplify`

{HypergeometricPFQ[{1/2,1/2,1/2},{1,1},-1],
Gamma[9/8]^2/(Gamma[5/4]^2*Gamma[7/8]^2),Pi/(Sqrt[2]*Gamma[5/8]^2*Gamma[7/8]^2)}

GammaSimplify/@%

Factor[FunctionExpand[%]]

FullSimplify[%]

N[%]