  We introduce families of rational functions that are biorthogonal with
respect to the $q$-hypergeometric distribution. A triplet of $q$-difference
operators $X$, $Y$, $Z$ is shown to play a role analogous to the pair of
bispectral operators of orthogonal polynomials. The recurrence relation and
difference equation take the form of generalized eigenvalue problems involving
the three operators. The algebra generated by $X$, $Y$, $Z$ is akin to the
algebras of Askey--Wilson type in the case of orthogonal polynomials. The
actions of these operators in three different basis are presented. Connections
with Wilson's ${}_{10}\phi_9$ biorthogonal rational functions are also
discussed.
