Vacuum polarization near asymptotically anti-de Sitter black holes in odd dimensions

Recently, Ba\~nados, Teitelboim and Zanelli obtained spherically symmetric black hole solutions in a particular class of Einstein--Lovelock gravity. We derive the propagator in an exact form for a conformal scalar field in the asymptotically anti-de Sitter black hole spacetime so as to study the quantum effects of the scalar fields. We treat the cases in odd dimensions in this paper. We calculate the vacuum expectation value of $\langle\varphi^2\rangle$ and show its dependence on the radial coordinate for the five-dimensional case as an example.