The Matérn Class of functions is given by $$k_M(r)=\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\frac{r\sqrt{2\nu}}{l}\right)^{\nu} K_{\nu}\left(\frac{r\sqrt{2\nu}}{l}\right)$$ where $r=\Vert x-x'\Vert$. The function $K_\nu$ is the Modified Bessel of the 2nd kind. And its spectral density is given by: $$S(x):=\frac{2^D\pi^{\frac{D}{2}}\Gamma(\nu+D/2)(2\nu)^{\nu}}{\Gamma(\nu)l^{2\nu}}\left(\frac{2\nu}{l^2}+4\pi^2 x^2\right)^{-(\nu+D/2)}$$, where $D$ is the dimension of $x$.
P.S.: For the definition of Matérn Covariance function and its spectral density, see William and Rasmussen's book on GP(Chapter 4)
However, in this R package, we have 
and they seem contradictory.
Which one is correct, and is there some bibliography for this? I've searched, but I couldn't find a link which I thought would trustworthy.
