  In a previous work a random matrix average for the Laguerre unitary ensemble,
generalising the generating function for the probability that an interval $
(0,s) $ at the hard edge contains $ k $ eigenvalues, was evaluated in terms of
a Painlev\'e V transcendent in $ \sigma $-form. However the boundary conditions
for the corresponding differential equation were not specified for the full
parameter space. Here this task is accomplished in general, and the obtained
functional form is compared against the most general small $ s $ behaviour of
the Painlev\'e V equation in $ \sigma $-form known from the work of Jimbo. An
analogous study is carried out for the the hard edge scaling limit of the
random matrix average, which we have previously evaluated in terms of a
Painlev\'e \IIId transcendent in $ \sigma $-form. An application of the latter
result is given to the rapid evaluation of a Hankel determinant appearing in a
recent work of Conrey, Rubinstein and Snaith relating to the derivative of the
Riemann zeta function.
