  We characterize averages of $\prod_{l=1}^N|x - t_l|^{\alpha - 1}$ with
respect to the Selberg density, further contrained so that $t_l \in [0,x]$
$(l=1,...,q)$ and $t_l \in [x,1]$ $(l=q+1,...,N)$, in terms of a basis of
solutions of a particular Fuchsian matrix differential equation. By making use
of the Dotsenko-Fateev integrals, the explicit form of the connection matrix
from the Frobenius type power series basis to this basis is calculated, thus
allowing us to explicitly compute coefficients in the power series expansion of
the averages. From these we are able to compute power series for the marginal
distributions of the $t_j$ $(j=1,...,N)$. In the case $q=0$ and $\alpha < 1$ we
compute the explicit leading order term in the $x \to 0$ asymptotic expansion,
which is of interest to the study of an effect known as singularity dominated
strong fluctuations. In the case $q=0$ and $\alpha \in \mathbb Z^+$, and with
the absolute values removed, the average is a polynomial, and we demonstrate
that its zeros are highly structured.
