A Generalization of Euler’s Hypergeometric Transformation

Euler’s two-term transformation formula for the Gauss hypergeometric function 2F1 is extended to hypergeometric functions of higher order. The generalized two-term transformation follows from the theory of Fuchsian differential equations with accessory parameters, although it also has a combinatorial proof. Unusually, it constrains the hypergeometric function parameters algebraically and not linearly. Its consequences for hypergeometric summation are explored. It has as corollary a summation formula of Slater, from which new one-term 3F2(1) and 2F1(−1) evaluations are derived by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new one-term evaluations of 2F1(−1) with linearly constrained parameters are also derived. It is shown that unlike Euler’s transformation, Pfaff’s transformation does not extend naturally to hypergeometric functions of higher order. 1991 Mathematics Subject Classification. Primary 33C20; Secondary 33C05, 34Mxx. Partially supported by NSF grant PHY-0099484.