Certain transformations and special values of hypergeometric functions over finite fields
type: Item
instance of: scholarly article
title: Certain transformations and special values of hypergeometric functions over finite fields
zbMATH Open document ID: 1490.33002
author: Rupam Barman
author: Mohit Tripathi
published in: The Ramanujan Journal
publication date: 2022-03-21T00:00:00Z
review text: \textit{J. Greene} [Trans. Am. Math. Soc. 301, 77--101 (1987; Zbl 0629.12017)] introduced hypergeometric functions over finite fields, which are also known as Gaussian hypergeometric series, and in particular deduced certain special values of \(_2F_1\) and \(_3F_2\) hypergeometric functions. Since then many special values of hypergeometric functions over finite fields are obtained using different techniques. Most of the known results give values of \(_2F_1\) and \(_3F_2\) hypergeometric functions containing trivial, quadratic and quartic characters as parameters. Finding values of Gaussian hypergeometric series containing arbitrary characters at specific values of the argument is a difficult problem and only a few such values are known till now. It is remarkable that special values of Gaussian hypergeometric series play an important role in solving many old conjectures and supercongruences. \textit{K. Ono} [Trans. Am. Math. Soc. 350, No. 3, 1205--1223 (1998; Zbl 0910.11054)] explicitly evaluated special values of \(_2F_1\) and \(_3F_2\) hypergeometric functions containing trivial and quadratic characters by counting points on elliptic curves with complex multiplication over \({\mathbb F}_p\). By relating the values of \(_3F_2\) hypergeometric functions obtained by Ono to squares of certain \(_2F_1\) hypergeometric functions, Kalita evaluated special values of squares of certain \(_2F_1\) hypergeometric functions containing trivial and quadratic characters. In the current work, Barman and Tripathi find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums, they evaluate certain character sums to establish these transformations. They then use their new transformation formulas to find several special values of Gaussian hypergeometric series including special values of \(_2F_1\) hypergeometric functions containing arbitrary characters at certain arguments. Certain special values of \…
reviewed by: Arman Shamsi Zargar
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 33C20
zbMATH DE Number: 7493917
zbMATH Keywords: hypergeometric series
zbMATH Keywords: Gauss and Jacobi sums
zbMATH Keywords: hypergeometric functions over finite fields
MaRDI profile type: Publication
full work available at URL: s11139-021-00410-1
OpenAlex ID: W3161098533
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DOI: 10.1007/S11139-021-00410-1
description: scientific article; zbMATH DE number 7493917