  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.
