  The functional structure of celestial amplitudes as constrained by Poincar\'e
symmetry is investigated in $2,3,$ and $4$-point cases for massless external
particles of various spin, as well as massive external scalars. Functional
constraints and recurrence relations are found (akin to the findings in
arXiv:1901.01622) that must be obeyed by the respective permissible correlator
structures and function coefficients. In specific three-point cases involving
massive scalars the resulting recurrence relations can be solved, e.g.
reproducing purely from symmetry a three-point function coefficient known in
the literature. Additionally, as a byproduct of the analysis, the three-point
function coefficient for gluons in Minkowski signature is obtained from an
amplitude map to the celestial sphere.
