Ramanujan 6-10-8 Identity

Author

Eric W. Weisstein
September 16, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Ramanujan6-10-8Identity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Ramanujan6-10-8Identity.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identity

Expand[Subtract@@{
64((a+b+c)^6+(b+c+d)^6-(c+d+a)^6-(d+a+b)^6+(a-d)^6-(b-c)^6)((a+b+c)^10+(b+c+d)^10-(c+d+a)^10-(d+a+b)^10+(a-d)^10-(b-c)^10),45((a+b+c)^8+(b+c+d)^8-(c+d+a)^8-(d+a+b)^8+(a-d)^8-(b-c)^8)^2
}/.d->b c/a]

Subtract@@{64((a x+(b+c)y)^6 +(b x-(a+c)y)^6 +(c x-(a-b)y)^6 -(a x-(b+c)y)^6
-(b x+(a+c)y)^6 -(c x+(a-b)y)^6)
((a x+(b+c)y)^10+(b x-(a+c)y)^10+(c x-(a-b)y)^10-(a x-(b+c)y)^10-(b x+(a+c)y)^10-(c x+(a-b)y)^10),
45((a x+(b+c)y)^8 +(b x-(a+c)y)^8 +(c x-(a-b)y)^8 -(a x-(b+c)y)^8
-(b x+(a+c)y)^8 -(c x+(a-b)y)^8)^2}/.c->a+b//Expand