Hirschhorn 3-7-5 Identity

Author

Eric W. Weisstein
September 16, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Hirschhorn3-7-5Identity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Hirschhorn3-7-5Identity.html.

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

Identity

Subtract@@{25((b+c+d)^3+(a-d)^3-(a+b+c)^3-(c+d+a)^3-(b-c)^3+(d+a+b)^3)((b+c+d)^7+(a-d)^7-(a+b+c)^7-(c+d+a)^7-(b-c)^7+(d+a+b)^7),21((b+c+d)^5+(a-d)^5-(a+b+c)^5-(c+d+a)^5-(b-c)^5+(d+a+b)^5)^2}/.d->b c/a//Expand

Subtract@@{25((a x+(b+c)y)^3+(b x-(a+c)y)^3-(c x-(a-b)y)^3-(a x-(b+c)y)^3-(b x+(a+c)y)^3+(c x+(a-b)y)^3)((a x+(b+c)y)^7+(b x-(a+c)y)^7-(c x-(a-b)y)^7-(a x-(b+c)y)^7-(b x+(a+c)y)^7+(c x+(a-b)y)^7),21((a x+(b+c)y)^5+(b x-(a+c)y)^5-(c x-(a-b)y)^5-(a x-(b+c)y)^5-(b x+(a+c)y)^5+(c x+(a-b)y)^5)^2}/.c->a+b//Expand

Subtract@@{25(p^3+q^3-(p+q)^3-r^3-s^3+(r+s)^3)(p^7+q^7-(p+q)^7-r^7-s^7+(r+s)^7)-21(p^5+q^5-(p+q)^5-r^5-s^5+(r+s)^5)^2,-525p q(p+q)r s(r+s)(p^2+p q+q^2-r^2-r s-s^2)^2}//Expand