Differentiability class of Matern function (based on Modified Bessel Function of second kind)

Question

I am working on some techniques using the Matérn covariance function:

$h(r) = \frac{2^{1-\nu}}{\Gamma(\nu)}\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)^\nu K_\nu\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)$ with $r\in\mathbb{R}^+$ and $\nu\in\mathbb{R}^+$.

For some reasons, I am looking at the continuity properties of this function wrt to $\nu$. By studying the first and second derivatives of it wrt $r$. I can show that these derivatives are continuous on $\mathbb{R}^+$ if $\nu>1$.

The idea now is to find the full continuity properties wrt to $\nu$. For doing this I am looking on some recurrence relations on the derivatives of $K_\nu(x)$ or $x^\nu K_\nu(x)$ wrt $r$.

Answer 1

Finally I have found in Stein 1999 - Interpolation Spatial Data - Some Theory for Kriging that the Matérn function is $2m$ times differentiable if and only if $\nu>m$ with $m\in\mathbb{N}^*$.

Answer 2

For $\text{Re}(x) > 0$, $$ K_\nu(x) = \int_0^\infty \exp(-x \cosh(t)) \cosh(\nu t)\; dt $$ which is analytic with respect to $\nu$.