  In the context of conformal field theories in general space-time dimension,
we find all the possible singularities of the conformal blocks as functions of
the scaling dimension $\Delta$ of the exchanged operator. In particular, we
argue, using representation theory of parabolic Verma modules, that in odd
spacetime dimension the singularities are only simple poles. We discuss how to
use this information to write recursion relations that determine the conformal
blocks. We first recover the recursion relation introduced in 1307.6856 for
conformal blocks of external scalar operators. We then generalize this
recursion relation for the conformal blocks associated to the four point
function of three scalar and one vector operator. Finally we specialize to the
case in which the vector operator is a conserved current.
