A generalization of Euler’s hypergeometric transformation

Maier, Robert

doi:10.1090/S0002-9947-05-04045-6

A generalization of Euler’s hypergeometric transformation
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by Robert S. Maier PDF

Trans. Amer. Math. Soc. 358 (2006), 39-57 Request permission

Abstract:

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.

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