I'm looking at the following generalizations for sums of two cubes. $u^3+v^3=(u+v)(u^2-uv+v^2).$
$u^3+(u+q)^3+(u+r)^3+(u+q+r)^3+v^3+(v+q)^3+(v+r)^3+(v+q+r)^3 = \\2(u+v+q+r)(2u^2-2uv+2v^2+qu+qv+ru+rv+2q^2+2r^2+qr).$
$u^3+(u+p)^3+(u+q)^3+(u+r)^3+(u+p+q)^3+(u+p+r)^3+(u+q+r)^3+(u+p+q+r)^3\\+v^3+(v+p)^3+(v+q)^3+(v+r)^3+(v+p+q)^3+(v+p+r)^3+(v+q+r)^3+(v+p+q+r)^3 = \\4(u+v+p+q+r)(2u^2-2uv+2v^2+pu+pv+qu+qv+ru+rv+2p^2+2q^2+2r^2+pq+pr+qr).$
These identities do generalize in this direction. It seems to me that identities this natural should be known or at least talked about. Is there any known literature that discusses these identities and the generalization in any detail?