  We consider the ratio of two Gauss hypergeometric functions with real
parameters shifted by arbitrary integers. We find a formula for the jump of
this ratio over the branch cut in terms of a real hypergeometric polynomial,
the beta density and the absolute value of the Gauss hypergeometric function.
This allows us to construct explicit integral representations for such ratio
when the asymptotic behaviour at unity is mild and the denominator does not
vanish. Multiplying the ratio by the aforementioned polynomial, we obtain a
function that belongs to a generalized Nevanlinna class for arbitrary values of
real parameters. We give an in-depth analysis of a particular case known as the
Gauss ratio. Furthermore, we establish a few general facts relating generalized
Nevanlinna classes to Jacobi and Stieltjes continued fractions, as well as to
factorization formulae for these classes. The results are illustrated with a
large number of examples.
