Evaluations of hypergeometric functions over finite fields

Ron Evans; John Greene

doi:10.32917/hmj/1249046338

Evaluations of hypergeometric functions over finite fields

Ron Evans, John Greene

Mathematics & Statistics

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

16 Scopus citations

Abstract

We prove two general formulas for a two-parameter family of hypergeometric ₃F₂(z) functions over a finite field F_q, where q is a power of an odd prime. Each formula evaluates a ₃F₂ in terms of a ₂F₁ over F_q2 As applications, we evaluate infinite one-parameter families of ₃F₂(1/4) and ₃F₂(-1), thereby extending results of J. Greene-D. Stanton and K. Ono, who gave evaluations in special cases.

Original language	English (US)
Pages (from-to)	217-235
Number of pages	19
Journal	Hiroshima Mathematical Journal
Volume	39
Issue number	2
DOIs	https://doi.org/10.32917/hmj/1249046338
State	Published - 2009

Bibliographical note

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Copyright 2020 Elsevier B.V., All rights reserved.

Keywords

Davenport-Hasse formulas
Gauss sums
Hypergeometric functions over finite fields
Jacobi sums
Lifted characters
Stickelberger's congruence

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10.32917/hmj/1249046338

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abstract = "We prove two general formulas for a two-parameter family of hypergeometric 3F2(z) functions over a finite field Fq, where q is a power of an odd prime. Each formula evaluates a 3F2 in terms of a 2F1 over Fq2 As applications, we evaluate infinite one-parameter families of 3F2(1/4) and 3F2(-1), thereby extending results of J. Greene-D. Stanton and K. Ono, who gave evaluations in special cases.",

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AB - We prove two general formulas for a two-parameter family of hypergeometric 3F2(z) functions over a finite field Fq, where q is a power of an odd prime. Each formula evaluates a 3F2 in terms of a 2F1 over Fq2 As applications, we evaluate infinite one-parameter families of 3F2(1/4) and 3F2(-1), thereby extending results of J. Greene-D. Stanton and K. Ono, who gave evaluations in special cases.

KW - Davenport-Hasse formulas

KW - Gauss sums

KW - Hypergeometric functions over finite fields

KW - Jacobi sums

KW - Lifted characters

KW - Stickelberger's congruence

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Evaluations of hypergeometric functions over finite fields

Abstract

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