I need your help. Given $K_\nu(x)$ the modified Bessel function of the second type (https://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html), what do you know about \begin{align*} f_\nu(x):=\vert x \vert^\nu K_\nu(\vert x \vert) \end{align*} ? For example I'd like to know its regularity, its differential equation, does it solve a Bessel's differential equation or some pde involving $\partial_\nu$ and $\partial_x$ ? From my simulation $f_\nu(\cdot) \in L^1 \cap L^\infty (\mathbb R)$. My focus is on $x \in \mathbb R$ and $\nu \in \mathbb R$, maybe for Bessel's equation we need $\nu \geq 2$.
Thanks to everybody!