Hartman-Watson distribution and hyperbolic-like heat kernels

We relate Gruet's formula for the heat kernel on real hyperbolic spaces to the commonly used one derived from Millson induction and distinguishing the parity of the dimensions. The bridge between both formulas is settled by Yor's result on the joint distribution of a Brownian motion and of its exponential functional at fixed time. This result allows further to relate Gruet's formula with real parameter to the heat kernel of the hyperbolic Jacobi operator and to derive a new integral representation for the heat kernel of the Maass Laplacian. When applied to harmonic AN groups (known also as Damek-Ricci spaces), Yor's result yields also a new integral representation of their corresponding heat kernels through the modified Bessel function of the second kind and the Hartman-Watson distribution. This newly obtained formula has the merit to unify both existing formulas in the same way Gruet's formula does for real hyperbolic spaces.