  Suppose that $\Gamma$ is a continuous and self-adjoint Hankel operator on
$L^2(0, \infty)$ and that $Lf=-(d/dx(a(x)df/dx))+b(x)f(x)$ with $a(0)=0$. If
$a$ and $b$ are both quadratic, hyperbolic or trigonometric functions, and
$\phi$ satisfies a suitable form of Gauss's hypergeometric equation, or the
confluent hypergeometric equation, then $L\Gamma =\Gamma L$. The paper
catalogues the commuting pairs $\Gamma$ and $L$, including important cases in
random matrix theory. There are also results proving rapid decay of the
singular numbers of Hankel integral operators with kernels that are analytic
and of exponential decay in the right half plane.
