Semigroup asymptotics, Funk-Hecke identity and the Gegenbauer coefficients associated with the spherical Laplacian

A trace formulation of the Maclaurin spectral coefficients of the Schwartzian kernel of functions of the spherical Laplacian is given. A class of polynomials Pv/l (X) (l >_ 0, v > -1/2) linking to the classical Gegenbauer polynomials through a differential-spectral identity is introduced and its connection to the above spectral coefficients and their asymptotics analysed. The paper discusses some applications of these ideas combined with the Funk-Hecke identity and semigroup techniques to geometric and variational-energy inequalities on the sphere and presents some examples.