As regards modified Bessel function of the first and second kind, in some cases the following relations hold:
$$I_{-n} (x) = I_{n} (x)\\ K_{-n} (x) = K_{n} (x)$$
- Under what conditions on $n$ these relations are true?
- Is there a site/document showing them?
My attempt: only as regards $K_{n} (x)$, I considered its integral definition:
$$K_{n} (x) = \int_0^{\infty} e^{-x \cosh t} \cosh (\alpha t) \mathrm{d}t, \ \mathrm{Re[x]} > 0$$
where trivially $\cosh(\alpha t) = \cosh(- \alpha t)$, being hyperbolic cosine an even function.
I am struggling to find a homogeneous source for all these types of functions. This one and this one don't mention at all the fact that $n$ should be integer or not. This one (page 19) shows the relation, but nothing is mentioned about $n$.
As regards the Bessel functions of the first and second kind:
$$J_{-n} (x) = (-1)^{n} J_{n} (x)\\ N_{-n} (x) = (-1)^{n} N_{n} (x)$$
the relations hold only for integer $n$, as stated in Wikipedia, respectively for $J_{n} (x)$ and $N_{n} (x)$.
This page shows that:
$$I_{-n} (x) = I_n (x)$$
only if $n$ is an integer. More in general, instead
$$I_{-\nu} (x) = I_{\nu} (x) + \frac{2}{\pi} \sin(\nu \pi) K_{\nu} (x)$$
The second relation has no limitations. It is valid for any $\nu$, not only the integer ones:
$$K_{-\nu} (x) = K_{\nu} (x)$$
In fact:
$$K_{\nu} (x) = \pi \frac{I_{-\nu} (x) - I_{\nu} (x)}{2 \sin(\pi \nu)}$$
and
$$K_{-\nu} (x) = \pi \frac{I_{\nu} (x) - I_{-\nu} (x)}{- 2 \sin(\pi \nu)} = \pi \frac{I_{-\nu} (x) - I_{\nu} (x)}{ 2 \sin(\pi \nu)} = K_{\nu} (x)$$