On Conformal Block, Crossing Kernel and Multi-variable Hypergeometric Functions

In this note, we present an alternative representation of the conformal block with external scalars in general spacetime dimensions in terms of a finite summation over Appell fourth hypergeometric function ${\bf{F}}_4$. We also construct its generalization to the non-local primary exchange operator with continuous spin and its corresponding Mellin representation which are relevant for Lorentzian spacetime. Using these results we apply the Lorentzian inversion formula to compute so-called crossing kernel in general spacetime dimensions, the resultant expression can be written as a double infinite summation over certain Kamp\~{e} de F\~{e}riet hypergeometric functions with the correct double trace operator singularity structures. We also include some complementary computations in AdS space, demonstrating the orthogonality of conformal blocks and performing the decompositions.