  It is shown that the classical quadratic and cubic transformation identities
satisfied by the hypergeometric function ${}_3F_2$ can be extended to include
additional parameter pairs, which differ by integers. In the extended
identities, which involve hypergeometric functions of arbitrarily high order,
the added parameters are nonlinearly constrained: in the quadratic case, they
are the negated roots of certain orthogonal polynomials of a discrete argument
(dual Hahn and Racah ones). Specializations and applications of the extended
identities are given, including an extension of Whipple's identity relating
very well poised ${}_7F_6(1)$ series and balanced ${}_4F_3(1)$ series, and
extensions of other summation identities.
