A spectral identity on Jacobi polynomials and its analytic implications

The Jacobi coefficients c`j (; ) (1 j `, ; > 1) are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the jacobi polynomials P(;) k (k 0; ; > 1) into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed and a direct trace interpretation of the Maclaurin coefficients is presented.