Extensions of the classical transformations of the hypergeometric function F 2 3

It is shown that the classical quadratic and cubic transformation identities satisfied by the hypergeometric function F 2 3 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 F 6 7 ( 1 ) series and balanced F 3 4 ( 1 ) series, and extensions of other summation identities.