We propose the Multiple reflection expansion as a tool for the calculation
of heat kernel coefficients. As an example, we give the coefficients for a
sphere as a finite sum over reflections, obtaining as a byproduct, a relati
on between the coefficients for Dirichlet and Neumann boundary conditions.
Further, we calculate the heat kernel coefficients for the most general mat
ching conditions on the surface of a sphere, including those cases correspo
nding to the presence of delta and delta prime background potentials. In th
e latter case, the multiple reflection expansion is shown to be nonconverge
nt.