A quadratic hypergeometric 2F1 transformation over finite fields

Ron Evans; John Greene

doi:10.1090/proc/13303

A quadratic hypergeometric ₂F₁ transformation over finite fields

Ron Evans
, John Greene

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric₂F₁ functions over a finite field F_q. We prove this conjecture in Theorem 2. The proof depends on a new linear transformation formula for pseudo-hypergeometric functions over F_q. Theorem 2 is then applied to give an elegant new transformation formula (Theorem 3) for₂F₁ functions over finite fields.

Original language	English (US)
Pages (from-to)	1071-1076
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	145
Issue number	3
DOIs	https://doi.org/10.1090/proc/13303
State	Published - 2017

Bibliographical note

Publisher Copyright:
© 2016 American Mathematical Society.

Keywords

Gauss sums
Hasse–Davenport relation
Hypergeometric F functions over finite fields
Jacobi sums
Pseudo-hypergeometric functions
Quadratic transformations

Publisher link

10.1090/proc/13303

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abstract = "In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric2F1 functions over a finite field Fq. We prove this conjecture in Theorem 2. The proof depends on a new linear transformation formula for pseudo-hypergeometric functions over Fq. Theorem 2 is then applied to give an elegant new transformation formula (Theorem 3) for2F1 functions over finite fields.",

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AB - In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric2F1 functions over a finite field Fq. We prove this conjecture in Theorem 2. The proof depends on a new linear transformation formula for pseudo-hypergeometric functions over Fq. Theorem 2 is then applied to give an elegant new transformation formula (Theorem 3) for2F1 functions over finite fields.

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KW - Hasse–Davenport relation

KW - Hypergeometric F functions over finite fields

KW - Jacobi sums

KW - Pseudo-hypergeometric functions

KW - Quadratic transformations

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