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A generalization of Euler's hypergeometric transformation


Author: Robert S. Maier
Journal: Trans. Amer. Math. Soc. 358 (2006), 39-57
MSC (2000): Primary 33C20; Secondary 33C05, 34Mxx
DOI: https://doi.org/10.1090/S0002-9947-05-04045-6
Published electronically: August 25, 2005
MathSciNet review: 2171222
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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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Additional Information

Robert S. Maier
Affiliation: Departments of Mathematics and Physics, University of Arizona, Tucson, Arizona 85721
Email: rsm@math.arizona.edu

DOI: https://doi.org/10.1090/S0002-9947-05-04045-6
Received by editor(s): April 11, 2003
Published electronically: August 25, 2005
Additional Notes: This work was partially supported by NSF grant PHY-0099484.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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