  The heat kernel expansion on even-dimensional hyperbolic spaces is asymptotic
at both short and long times, with interestingly different Borel properties for
these short and long time expansions. Resummations in terms of incomplete gamma
functions provide accurate extrapolations and analytic continuations, relating
the heat kernel to the Schrodinger kernel, and the heat kernel on hyperbolic
space to the heat kernel on spheres. For the diagonal heat kernel there is also
a duality between short and long times which mixes the scalar and spinor heat
kernels.
