  In this article, we consider flat and curved Riemannian symmetric spaces in
the complex case and we study their basic integral kernels, in potential and
spherical analysis: heat, Newton, Poisson kernels and spherical functions, i.e.
the kernel of the spherical Fourier transform.
  We introduce and exploit a simple new method of construction of these
$W$-invariant kernels by alternating sum formulas. We then use the alternating
sum representation of these kernels to obtain their asymptotic behavior. We
apply our results to the Dyson Brownian Motion on $R^d$.
