Asymptotic identities for Jacobi polynomials via spectral geometry of rank-one symmetric spaces

Radial eigenfunctions of the Laplace-Beltrami operator on compact rank-one symmetric spaces may be expressed in terms of Jacobi polynomials. We use this fact to prove an identity for Jacobi polynomials which is inspired by results of Minakshisundaram-Pleijel and Zelditch on the Fourier coefficients of a smooth measure supported on a compact submanifold of a compact Riemannian manifold.