  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.
