  An eigenfunction expansion method involving hypergeometric functions is used
to solve the partial differential equation governing the transport of radiation
in an X-ray pulsar accretion column containing a radiative shock. The procedure
yields the exact solution for the Green's function, which describes the
scattering of monochromatic radiation injected into the column from a source
located near the surface of the star. Collisions between the injected photons
and the infalling electrons cause the radiation to gain energy as it diffuses
through the gas and gradually escapes by passing through the walls of the
column. The presence of the shock enhances the energization of the radiation
and creates a power-law spectrum at high energies, which is typical for a Fermi
process. The analytical solution for the Green's function provides important
physical insight into the spectral formation process in X-ray pulsars, and it
also has direct relevance for the interpretation of spectral data for these
sources. Additional interesting mathematical aspects of the problem include the
establishment of a closed-form expression for the quadratic normalization
integrals of the orthogonal eigenfunctions, and the derivation of a new
summation formula involving products of hypergeometric functions. By taking
various limits of the general expressions, we also develop new linear and
bilinear generating functions for the Jacobi polynomials.
