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A quadratic hypergeometric $ {}_2F_1$ transformation over finite fields

Authors: Ron Evans and John Greene
Journal: Proc. Amer. Math. Soc. 145 (2017), 1071-1076
MSC (2010): Primary 11T24, 33C05
Published electronically: October 18, 2016
MathSciNet review: 3589307
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Abstract: In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric $ {}_2\hspace {-1pt}F_1$ functions over a finite field $ \mathbb{F}_q$. We prove this conjecture in Theorem 2. The proof depends on a new linear transformation formula for pseudo-hypergeometric functions over $ \mathbb{F}_q$. Theorem 2 is then applied to give an elegant new transformation formula (Theorem 3) for $ {}_2\hspace {-1pt}F_1$ functions over finite fields.

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Additional Information

Ron Evans
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

John Greene
Affiliation: Department of Mathematics and Statistics, University of Minnesota–Duluth, Duluth, Minnesota 55812

Keywords: Hypergeometric ${}_2F_1$ functions over finite fields, Gauss sums, Jacobi sums, pseudo-hypergeometric functions, quadratic transformations, Hasse--Davenport relation
Received by editor(s): October 30, 2015
Received by editor(s) in revised form: November 5, 2015, November 18, 2015, and May 18, 2016
Published electronically: October 18, 2016
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society

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