A Spectral Identity on Jacobi Polynomials and its Analytic Implications

Abstract

AbstractThe Jacobi coefficientsare 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 polynomialsinto 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.