Hartman-Watson distribution and hyperbolic-like heat kernels

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