  Elastic channels are an important component of many soft matter systems, in
which hydrodynamic interactions with confining membranes determine the behavior
of particles in flow. In this work, we derive analytical expressions for the
Green's functions associated to a point-force (Stokeslet) directed parallel or
perpendicular to the axis of an elastic cylindrical channel exhibiting
resistance against shear and bending. We then compute the leading order self-
and pair mobility functions of particles on the cylinder axis, finding that the
mobilities are primarily determined by membrane shear and that bending does not
play a significant role. In the quasi-steady limit of vanishing frequency, the
particle self- and pair mobilities near a no-slip hard cylinder are recovered
only if the membrane possesses a non-vanishing shear rigidity. We further
compute the membrane deformation, finding that deformation is generally more
pronounced in the axial (radial) directions, for the motion along
(perpendicular) to the cylinder centerline, respectively. Our analytical
calculations for the Green's functions in an elastic cylinder can serve as a
fundamental building block for future studies and are verified by fully
resolved boundary integral simulations where a very good agreement is obtained.
