Hypergeometric polynomials over finite fields
type: Item
instance of: scholarly article
title: Hypergeometric polynomials over finite fields
zbMATH Open document ID: 0949.11060
author: Masao Koike
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1999-10-10T00:00:00Z
review text: Let \(p\geq 5\) be a prime. For elements \(a,b,c\) of the finite field \(\mathbb{F}_p\) the hypergeometric polynomial \(F(a,b,c;x)\) is defined by \[ F(a,b,c;x)=\sum_{n=0} \frac{(a)_n(b)_n}{(1)_n(c)_n}x^n, \] where \((a)_n=a(a+1)\cdots (a+n-1)\) and the sum stops as soon as the numerator vanishes under the assumption that the denominator does not vanish before the numerator vanishes. Moreover, for \(\varepsilon, \varepsilon'\in \{-1,1\}\) let \[ F_{\varepsilon,\varepsilon'}(x)=\prod_a(x-a), \] where \(a\) runs over all elements \(a\in \mathbb{F}_p^*\) satisfying \(\left(\frac{a}{p}\right)=\varepsilon\) and \((\frac{1-a}{p})=\varepsilon',\) where \((\frac{\cdot}{p})\) denotes the Legendre symbol. \textit{T. Honda} [Algebraic geometry, Roma 1979, Symp. Math. 24, 169-204 (1981; Zbl 0464.12013)] evaluated \(F_{-1,\pm 1}(x)\) in terms of hypergeometric polynomials. In the paper under review the author evaluates the remaining polynomials \(F_{1,\pm 1}(x)\): \[ F_{1,-1}(x)=a_{1,-1}^{(p)}F\left(\frac{1}{4},\frac{3}{4},\frac{1}{2};x\right)\text{ and } F_{1,1}(x)=a_{1,1}^{(p)}F\left(\frac{3}{4},\frac{5}{4},\frac{3}{2};x\right), \] where \[ a_{1,-1}^{(p)}=\begin{cases} 1 &\text{if }p\equiv 1\bmod 4,\\ -2&\text{if }p\equiv 3\bmod 4,\end{cases}\quad\text{ and }\quad a_{1,1}^{(p)}=\begin{cases} 1 &\text{if }p\equiv 1\bmod 4,\\ -1/2&\text{if }p\equiv 3\bmod 4.\end{cases}. \]
reviewed by: Arne Winterhof
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 33E50
zbMATH DE Number: 1321875
zbMATH Keywords: hypergeometric polynomials
zbMATH Keywords: finite fields
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224854
OpenAlex ID: W1970810642
cites work: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
cites work: Q3915082
cites work: CLASS NUMBER OF A DEFINITE QUATERNION WITH PRIM DISCRIMINANT
cites work: Q4767396
DOI: 10.2748/TMJ/1178224854
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description: scientific article; zbMATH DE number 1321875