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A multipoint conformal block chain in $d$ dimensions

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.