  A family of random variables $\mathbf{X}(s)$, depending on a real parameter
$s>-\frac{1}{2}$, appears in the asymptotics of the joint moments of
characteristic polynomials of random unitary matrices and their derivatives, in
the ergodic decomposition of the Hua-Pickrell measures and conjecturally in the
asymptotics of the joint moments of Hardy's function and its derivative. Our
first main result establishes a connection between the characteristic function
of $\mathbf{X}(s)$ and the $\sigma$-Painlev\'e III' equation in the full range
of parameter values $s>-\frac{1}{2}$. Our second main result gives the first
explicit expression for the density and all the complex moments of the absolute
value of $\mathbf{X}(s)$ for integer values of $s$. Finally, we establish an
analogous connection to another special case of the $\sigma$-Painlev\'e III'
equation for the Laplace transform of the sum of the inverse points of the
Bessel point process.
