Singular expansion of the wave kernel and harmonic sums on Riemannian symmetric spaces of the non-compact type

The Mellin transform assigned to the convolution Poisson kernel on higher rank Riemannian symmetric spaces of the non-compact type is equal to the wave kernel. This makes it possible to determine the poles and to deduce the singular expansion of this kernel by using the zeta function techniques on compact and non-compact manifolds. As a consequence, we studied the harmonic sums associated with the wave kernel. In particular, we derived its asymptotic expansion near $0$ according to the Mellin-converse correspondence rule.