  This paper studies the properties of the probability density function
$p_{\alpha,\nu, n}(\mathbf{x})$ of the $n$-variate generalized Linnik
distribution whose characteristic function
$\varphi_{\alpha,\nu,n}(\boldsymbol{t})$ is given by
\varphi_{\alpha,\nu,n}(\boldsymbol{t})=\frac{1}
{(1+\Vert\boldsymbol{t}\Vert^{\alpha})^{\nu}}, \alpha\in (0,2], \nu>0,
\boldsymbol{t}\in \mathbb{R}^n, where $\Vert\boldsymbol{t}\Vert$ is the
Euclidean norm of $\boldsymbol{t}\in\mathbb{R}^n$. Integral representations of
$p_{\alpha,\nu, n}(\mathbf{x})$ are obtained and used to derive the asymptotic
expansions of $p_{\alpha,\nu, n}(\mathbf{x})$ when $\Vert\mathbf{x}\Vert\to 0$
and $\Vert\mathbf{x}\Vert\to \infty$ respectively. It is shown that under
certain conditions which are arithmetic in nature, $p_{\alpha,\nu,
n}(\mathbf{x})$ can be represented in terms of entire functions.
