Extensions of the Classical Transformations of the Hypergeometric Function ₃F₂

It is shown that the classical quadratic and cubic transformation identities satisfied by the hypergeometric function ₃F₂ 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 ₇F₆(1) series and balanced ₄F₃(1) series, and extensions of other summation identities.

MSC Codes Primary 33C20, 33C45