Analytic and asymptotic properties of multivariate Linnik's distribution

The paper deals with properties of $k$-variate ($k>2$) Linnik's distribution defined by the characteristic function
$$\varphi_{\alpha k}(t)=1/(1+|t|^\alpha),\quad0<\alpha<2,\quad t\in\mathrm R^k,$$
where $|t|$ denotes Euclidean norm of vector $t\in\mathrm R^k$. This distribution is absolutely continuous with respect to the Lebesgue measure in $R^k$. Expansions of the density of the distribution into asymptotic and convergent series in powers of $|t|$, $|t|^\alpha$ are obtained. The forms of these expansions depend substantially on the arithmetical nature of the parameter $\alpha$.