  Conformal blocks play a central role in CFTs as the basic, theory-independent
building blocks. However, only limited results are available concerning
multipoint blocks associated with the global conformal group. In this paper, we
systematically work out the $d$-dimensional $n$-point global conformal blocks
(for arbitrary $d$ and $n$) for external and exchanged scalar operators in the
so-called comb channel. We use kinematic aspects of holography and previously
worked out higher-point AdS propagator identities to first obtain the geodesic
diagram representation for the $(n+2)$-point block. Subsequently, upon taking a
particular double-OPE limit, we obtain an explicit power series expansion for
the $n$-point block expressed in terms of powers of conformal cross-ratios.
Interestingly, the expansion coefficient is written entirely in terms of
Pochhammer symbols and $(n-4)$ factors of the generalized hypergeometric
function ${}_3F_2$, for which we provide a holographic explanation. This
generalizes the results previously obtained in the literature for $n=4, 5$. We
verify the results explicitly in embedding space using conformal Casimir
equations.
