On Gegenbauer Polynomials and Coefficients $$\varvec{c^{\ell }_{j}(\nu )}$$ $$\varvec{(1\le j\le \ell }$$ , $$\varvec{\nu >-1/2)}$$.

The Gegenbauer coefficients $$c^{\ell }_{j}(\nu )$$ ( $$1\le j\le \ell $$ , $$\nu >-1/2$$ ) 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 $$C_{k}^{\nu }$$ ( $$k\ge 0, \nu >-1/2$$ ) into a spectral sum involving the powers of the eigenvalues of the associated Gegenbauer operator. We present explicit computations and various implications. [ABSTRACT FROM AUTHOR]