How to you prove that the Heat Kernel of the Sub-Laplacian is given by this formula?

Question

I been trying to show that the two integrals are equal but to no avail ,is my approach correct,I been looking at numerous literature and still could not see how those 2 integrals are equal,is one integral an approximation of another or are they equal? I been using hyperbolic functions identities all day and am truly stuck

I been looking at the papers 1) B Gaveau,Principle de Moindre action,propagation de la chaleur,et estimees sous,elliptiques sur certains groupes nilpotents 2) A Hulanicki,The Distribution of energy in the Brownian motion in the Guassian Field and analytic hypoellipticity of certain subellitpic operators

Here is my question,please see 2 images attached The Explicit formula for the Heat kernel of the Sub Laplacian

and Another Explicit formula for the Heat kernel of the Sub Laplacian which i need to prove

Answer 1

I think they're equal. You make the change of variables $\tau' = \tau \rho$ in the $K_\rho$ integral. You get one $1/\rho$ factor from the $\tau = \tau' / \rho$ in the denominator, and the other from $d\tau = d\tau' / \rho$. No hyperbolic identities needed.

Answer 2

I have solved the problem thanks to @Nate Eldrege comment and looking at the paper for a more general case of the question in the paper titled Semiclassical Limits of Heat Laplacians on h-heisenberg groups(https://www.mmnp-journal.org/articles/mmnp/pdf/2013/01/mmnp201381p132.pdf) Here is my attempt at the question, thank you again for the help

I was just wondering how would you evaluate the integral explicitly,I have already been using Wolfram Alpha and to no avail?