  Induced by three gluons symmetry, Mandelstam variables $s$, $t$, $u$
symmetric expressions are widely involved in collider physics, especially in
heavy quarkonium physics. In this work we study general form of $s$, $t$, $u$
symmetric polynomials, and find that they can be expressed as polynomials where
the symmetry is manifest. The general form is then used to simplify expressions
which asymptotically reduces the length of original expression to one-sixth.
Based on the general form, we reproduce the exact differential cross section of
$J/\psi$ hadron production at leading order in $v^2$ up to four unknown
constant numbers by simple analysis. Furthermore, we prove that differential
cross section at higher order in $v^2$ is proportional to that at leading
order. This proof explains the proportion relation at next-to-leading order in
$v^2$ found in previous work and generalizes it to all order.
