Is there a reference for these types of cubic identities?

Question

I'm looking at the following generalizations for sums of two cubes. $u^3+v^3=(u+v)(u^2-uv+v^2).$

$u^3+(u+r)^3+v^3+(v+r)^3=(u+v+r)(2u^2-2uv+2v^2+ru+rv+2r^2).$

$u^3+(u+q)^3+(u+r)^3+(u+q+r)^3+v^3+(v+q)^3+(v+r)^3+(v+q+r)^3 = \\2(u+v+q+r)(2u^2-2uv+2v^2+qu+qv+ru+rv+2q^2+2r^2+qr).$

$u^3+(u+p)^3+(u+q)^3+(u+r)^3+(u+p+q)^3+(u+p+r)^3+(u+q+r)^3+(u+p+q+r)^3\\+v^3+(v+p)^3+(v+q)^3+(v+r)^3+(v+p+q)^3+(v+p+r)^3+(v+q+r)^3+(v+p+q+r)^3 = \\4(u+v+p+q+r)(2u^2-2uv+2v^2+pu+pv+qu+qv+ru+rv+2p^2+2q^2+2r^2+pq+pr+qr).$

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?

Answer 1

Define $$ f(u, v) := u^3 + v^3 = (u+v)g(u, v),\quad g(u, v) := (u^2 -uv +v^2). $$ That is your first identity.

Note that if $\,z := u + v + r\,$ then $\, z = u+(v+r) = (u+r)+v.\,$ Thus, $$ f(u, v+r) + f(u+r, v) = u^3 + (v+r)^3 + (u+r)^3 + v^3 = \\ z\,g(u, v+r) + z\,g(u+r, v) = (u+r+v)(g(u, v+r) + g(u+r, v)). $$ That is your second identity.

Note that if $\,z := u + v + q + r\,$ then your third identity also has a factor of $\,z\,$ on the right side for similar reasons. This generalizes as you mentioned in your question. You asked:

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?

Natural is a matter of opinion, but these identities (except the first with a Wikipedia article) seem to have very limited application. This is probably why I think that there is very little literature references. On the positive side, your family of identities seems new, interesting and special. On that basis I have included your second identity as $\texttt{id3_5_2_3c}$ in my list of "Special Algebraic Identities" at my CSU home page.