  The operator product expansion (OPE) on the celestial sphere of conformal
primary gluons and gravitons is studied. Asymptotic symmetries imply recursion
relations between products of operators whose conformal weights differ by
half-integers. It is shown, for tree-level Einstein-Yang-Mills theory, that
these recursion relations are so constraining that they completely fix the
leading celestial OPE coefficients in terms of the Euler beta function. The
poles in the beta functions are associated with conformally soft currents.
