  We consider 5-point functions in conformal field theories in d > 2
dimensions. Using weight-shifting operators, we derive recursion relations
which allow for the computation of arbitrary conformal blocks appearing in
5-point functions of scalar operators, reducing them to a linear combination of
blocks with scalars exchanged. We additionally derive recursion relations for
the conformal blocks which appear when one of the external operators in the
5-point function has spin 1 or 2. Our results allow us to formulate positivity
constraints using 5-point functions which describe the expectation value of the
energy operator in bilocal states created by two scalars.
