  In a recent paper (Asci \textit{et al.}, 2008) it has been shown that certain
random continued fractions have a density which is a product of a beta density
and a hypergeometric function $_{2}F_{1}$. In the present paper we fully
exploit a formula due to Thomae (1879) in order to generalize substantially the
class of random continuous fractions with a density of the above form. This
involves the design of seven particular graphs. Infinite paths on them lead to
random continued fractions with an explicit distribution. A careful study about
the set of five real parameters leading to a beta-hypergeometric distribution
is required, relying on almost forgotten results mainly due to Felix Klein.
