A Spectral Identity on Jacobi Polynomials and its Analytic Implications

Source abstract

Abstract The Jacobi coefficients 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 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.