Explicit formulas of the heat kernel on the quaternionic projective spaces

We consider the heat equation on the quaternionic projective space P-n(H), and we establish two formulas for the heat kernel, a series expansion involving Jacobi polynomials, and an integral representation involving a theta-function. More precisely, using the quaternonic Hopf fibration and the explicit integral representation of the heat kernel on the complex projective space P2+1 (C) , as well as an integral representation of Jacobi polynomials in terms of Gegenbauer polynomials, we give an explicit integral representation of the heat kernel on the n-quaternionic projective space. We also establish an explicit series expansion of the heat kernel in terms of the Jacobi polynomials. Moreover, we derive an explicit formula of the heat kernel on the sphere S-4.