  This paper is devoted to an in deep investigation of the first fundamental
solution to the linear multi-dimensional space-time-fractional diffusion-wave
equation. This equation is obtained from the diffusion equation by replacing
the first order time-deri\-va\-ti\-ve by the Caputo fractional derivative of
order $\beta,\ 0 <\beta \leq 2$ and the Laplace operator by the fractional
Laplacian $(-\Delta)^{\frac\alpha 2}$ with $0<\alpha \leq 2$. First, a
representation of the fundamental solution in form of a Mellin-Barnes integral
is deduced by employing the technique of the Mellin integral transform. This
representation is then used for establishing of several subordination formulas
that connect the fundamental solutions for different values of the fractional
derivatives $\alpha$ and $\beta$. We also discuss some new cases of completely
monotone functions and probability density functions that are expressed in
terms of the Mittag-Leffler function, the Wright function, and the generalized
Wright function.
