  We study a technical problem arising from the spectral geometry of
noncommutative tori: the small time heat trace asymptotic associated to a
general second order elliptic operator. We extend the rearrangement operators
in the conformal case to the general setting using hypergeometric integrals
over Grassmannians. The main result is the explicit formula of the second heat
coefficient in terms of the coefficients. When specializing to examples in
conformal case, we not only recover results in previous works but also obtain
some extra functional relations whose validation provides experimental support
to the main results. At last, we verify the relations based on combinatorial
properties derived from the hypergeometric features.
