Based on Spiridonov's analysis of elliptic generalizations of the Gauss
hypergeometric function, we develop a common framework for 7-parameter families of
generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In
each case we derive the symmetries of the generalized hypergeometric function under the
Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the
appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous
relations using fundamental addition formulas for theta and sine functions. The top level
degenerations of the hyperbolic and trigonometric hypergeometric functions are identified
with Ruijsenaars' relativistic hypergeometric function and the Askey-Wilson function,
respectively. We show that the degeneration process yields various new and known identities
for hyperbolic and trigonometric special functions. We also describe an intimate connection
between the hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.