A family of random variables
$X\left(s\right)$,
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 (Assiotis et al. 2020), in the ergodic
decomposition of the Hua–Pickrell measures (Borodin and Olshanski 2001 and Qiu
2017), and conjecturally in the asymptotics of the joint moments of Hardy’s
function and its derivative (Hughes 2001 and Assiotis et al. 2020). Our first
main result establishes a connection between the characteristic function of
$X\left(s\right)$ and the
$\sigma $Painlevé
${\text{III}}^{\prime}$ 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
$X\left(s\right)$ for integer
values of
$s$.
Finally, we establish an analogous connection to another special case of the
$\sigma $Painlevé
${\text{III}}^{\prime}$
equation for the Laplace transform of the sum of the inverse points of the Bessel
point process.
