  We consider the complex $\nu$ plane structure of the associated Legendre
function of the second kind $Q^{-1/2-K}_{\nu}(\cosh\rho)$. We find that for any
noninteger value for $K$ $Q^{-1/2-K}_{\nu}(\cosh\rho)$ has an infinite number
of poles in the complex $\nu$ plane, but for any negative integer $K$ there are
no poles at all. For $K=0$ or any positive integer $K$ there is only a finite
number of poles, with there only being one single pole (at $\nu=0$) when $K=0$.
This pattern is characteristic of the exceptional points that appear in a wide
variety of physical contexts. However, unusually for theories with exceptional
points, $Q^{-1/2-K}_{\nu}(\cosh\rho)$ has an infinite number of them. Other
than in the $PT$-symmetry Jordan-block case, exceptional points usually occur
at complex values of parameters. While not being Jordan-block exceptional
points themselves, the exceptional points associated with the
$Q^{-1/2-K}_{\nu}(\cosh\rho)$ nonetheless occur at real values of $K$.
