  By using the generalized Abel-Plana formula, we derive a summation formula
for the series over the zeros of a combination of the associated Legendre
functions with respect to the degree. The summation formula for the series over
the zeros of the combination of the Bessel functions, previously discussed in
the literature, is obtained as a limiting case. As an application we evaluate
the Wightman function for a scalar field with general curvature coupling
parameter in the region between concentric spherical shells on background of
constant negative curvature space. For the Dirichlet boundary conditions the
corresponding mode-sum contains series over the zeros of the combination of the
associated Legendre functions. The application of the summation formula allows
us to present the Wightman function in the form of the sum of two integrals.
The first one corresponds to the Wightman function for the geometry of a single
spherical shell and the second one is induced by the presence of the second
shell. The boundary-induced part in the vacuum expectation value of the field
squared is investigated. For points away from the boundaries the corresponding
renormalization procedure is reduced to that for the boundary-free part.
