  We study the distribution of the smallest eigenvalue for certain classes of
positive-definite Hermitian random matrices, in the limit where the size of the
matrices becomes large. Their limit distributions can be expressed as Fredholm
determinants of integral operators associated to kernels built out of Meijer
$G$-functions or Wright's generalized Bessel functions. They generalize in a
natural way the hard edge Bessel kernel Fredholm determinant. We express the
logarithmic derivatives of the Fredholm determinants identically in terms of a
$2\times 2$ Riemann-Hilbert problem, and use this representation to obtain the
so-called large gap asymptotics.
