Are Hankel functions linearly independent for all real $m$? For $m \in Z$ , $J_m (x)$ and $Y_m (x)$ form a pair of linearly independent equations.
If $m$ is real and $m \not \in Z$ then solutions is given by $J_m (x)$ and $J_{-m} (x)$.
It's easy to find a reference for above statements. It surprisingly hard to find any article or book where independence of first and second Hankel functions is stated explicitly for all $m$. Are they independent for all real $m$? Could you kindly provide a reference for that statement?
Linear independence over $\mathbb{C}$ of $H^{(1)}_\nu(z)$ and $H^{(2)}_\nu(z)$ follows from linear independence of $J_\nu(z)$ and $Y_\nu(z)$ over $\mathbb{C}$, since $H^{(1)}_\nu(z) = J_\nu(z) + iY_\nu(z)$ and $H^{(2)}_\nu(z) = J_\nu(z) - iY_\nu(z)$ (see [1] section 3.6).
Indeed given $c_1, c_2 \in \mathbb{C}$ such that $c_1H^{(1)}_\nu(z) + c_2H^{(2)}_\nu(z) = 0$, we have $(c_1 + c_2)J_\nu(z) + i(c_1 - c_2)Y_\nu(z) = 0$.
By linear independence of $J_\nu(z)$ and $Y_\nu(z)$ over $\mathbb{C}$ we have $c_1 + c_2 = 0$ and $i(c_1 - c_2) = 0$, which means $c_1 = c_2 = 0$, as required.
Now to show that $J_\nu(z)$ and $Y_\nu(z)$ are linearly independent over $\mathbb{C}$ for any $\nu \in \mathbb{C}$.
When $\nu \not\in \mathbb{Z}$, the Neumann functions are defined by $$Y_\nu(z) = \frac{J_\nu(z)\cos(\nu\pi) - J_{-\nu}(z)}{\sin(\nu\pi)}$$ and for $n \in \mathbb{Z}$, we define $$Y_n(z) = \lim\limits_{\nu\to n}\frac{J_\nu(z)\cos(\nu\pi) - J_{-\nu}(z)}{\sin(\nu\pi)}$$ (see [1] section 3.54).
For any $\nu \in \mathbb{C}$, the Wronskian of $J_\nu$ and $J_{-\nu}$ is $W[J_\nu(z), J_{-\nu}(z)] = -\frac{2\sin(\nu\pi)}{\pi z}$ (see [1] section 3.12).
Then when $\nu \not\in \mathbb{Z}$, the Wronskian of $J_\nu$ and $Y_\nu$ is easily computed to be (by linearity of the Wronskian with respect to second argument, and using the definition for $Y_\nu$ above) $$W[J_\nu(z), Y_\nu(z)] = \frac{2}{\pi z}$$ and the result also holds when $\nu$ is an integer by continuity (see [1] section 3.63).
Hence $J_\nu$ and $Y_\nu$ are also linearly independent over $\mathbb{C}$.
It is also possible to show that $H^{(1)}_\nu(z)$ and $H^{(2)}_\nu(z)$ are linearly independent over $\mathbb{C}$ for any $\nu$ by computing the Wronskian $$W[H^{(1)}_\nu(z), H^{(2)}_\nu(z)] = -\frac{4i}{\pi z}$$ (see [1] section 3.63).
[1] Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press, viii, 804 S. (1922). ZBL48.0412.02.).