On a distinguished family of random variables and Painlevé equations

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