  We construct the coherent states of general order, $m$ for the ladder
operators, $c(m)$ and $c^\dagger(m)$, which act on rational deformations of the
harmonic oscillator. The position wavefunctions of the eigenvectors involve
type III Hermite exceptional orthogonal polynomials. We plot energy
expectations, time-dependent position probability densities for the coherent
states and for the even and odd cat states, Wigner functions, and Heisenberg
uncertainty relations. We find generally non-classical behaviour, with one
exception: there is a regime of large magnitude of the coherent state
parameter, $z$, where the otherwise indistinct position probability density
separates into $m+1$ distinct wavepackets oscillating and colliding in the
potential, forming interference fringes when they collide. The Mandel $Q$
parameter is calculated to find sub-Poissonian statistics, another indicator of
non-classical behaviour. We plot the position standard deviation and find
squeezing in many of the cases. We calculate the two-photon-number probability
density for the output state when the $m=4$, $\mu=-5$ coherent states (where
$\mu$ labels the lowest weight in the superposition) are placed on one arm of a
beamsplitter. We find that it does not factorize, again indicating
non-classical behaviour. Calculation of the linear entropy for this
beamsplitter output state shows significant entanglement, another non-classical
feature. We also construct linearized versions, $\tilde c(m)$, of the
annihilation operators and their coherent states and calculate the same
properties that we investigate for the coherent states. For these we find
similar behaviour to the $c(m)$ coherent states, at much smaller magnitudes of
$z$, but comparable average energies.
