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.

We considered the Mellin transform assigned to the convolution wave kernel associated to the Laplace-Beltrami operator on higher rank Riemannian symmetric spaces of the non-compact type. The occurrence of the analyticity strip of this transform can be deduced directly from the pointwise kernel estimates. Using the zeta function techniques, we established its meromorphic extension to the entire complex plane $\mathbb{C}$ with simple poles on the real line.