A generalization of Euler's hypergeometric transformation

Euler's transformation formula for the Gauss hypergeometric function ₂F ₁ is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly. Its consequences for hypergeometric summation are explored. It has as a corollary a summation formula of Slater. From this formula new one-term evaluations of ₂F ₁ (-1) and ₃F ₂(1) are derived by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new one-term evaluations of ₂F ₁(-1) with linearly constrained parameters are derived as well.