On Gegenbauer polynomials and coefficients clj(?) (1=j=l, ?>-1/2)

The Gegenbauer coefficients cjl(?)(1=j=l, ?>-1/2) appear in the Maclaurin expansion of the heat kernels on the n-sphere and the real projective n-space. In this note we show that these coefficients can be computed by transforming the higher order derivative formula for the Gegenbauer polynomials Ck?(k=0,?>-1/2) into a spectral sum involving the powers of the eigenvalues of the associated Gegenbauer operator. We present explicit computations and various implications.