Based on the idea of adiabatic expansion theory, we will present a new formula for the asymptotic expansion coefficients of every derivative of the heat kernel on a compact Riemannian manifold. It will be very useful for having systematic understanding of the coefficients, and, furthermore, by using only a basic knowledge of calculus added to the formula, one can describe them explicitly up to an arbitrarily high order.
"A formula for the heat kernel coefficients on Riemannian manifolds." Tohoku Math. J. (2) 72 (2) 261 - 282, 2020. https://doi.org/10.2748/tmj/1593136821