A generalization of Euler's hypergeometric transformation

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16 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)39-57
Number of pages19
JournalTransactions of the American Mathematical Society
Issue number1
StatePublished - Jan 2006

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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