  In the context of holographic conformal field theories (CFTs), a system of
linear partial differential equations was recently proposed to be the
higher-dimensional analog of the null-state equations in $d=2$ CFTs at large
central charge. Solving these equations in a near-lightcone expansion yields
solutions that match the minimal-twist multi-stress tensor contributions to a
heavy-light four-point correlator (or a thermal two-point correlator) computed
using holography, the conformal bootstrap, and other methods. This note
explores the exact solutions to these equations. We begin by observing that, in
an expansion in terms of the ratio between the heavy operator's dimension and
the central charge, the $d=2$ correlator involving the level-two degenerate
scalars at each order can be represented as a Bessel function; the resummation
yields the Virasoro vacuum block. We next observe a relation between the $d=2$
correlator and the $d=4$ near-lightcone correlator involving light scalars with
the same conformal dimension. The resummed $d=4$ correlator takes a simple form
in the complex frequency domain. Unlike the Virasoro vacuum block, the
resummation in $d=4$ leads to essential singularities. Similar expressions are
also obtained when the light scalar's dimension takes other finite values.
These CFT results correspond to a holographic computation with a spherical
black hole. In addition, using the differential equations, we demonstrate that
the correlators can be reconstructed via certain modes. In $d=2$, these modes
are related to the Virasoro algebra.
