Certain product formulas and values of Gaussian hypergeometric series
Certain product formulas and values of Gaussian hypergeometric series
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高斯超几何级数的某些乘积公式和值
DOI：
10.1007/s40993-020-00203-3
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发表时间：
2020
影响因子：
0.8
通讯作者：
Rupam Barman
中科院分区：
文献类型：
--
作者：
M. Tripathi;Rupam Barman
在这篇文章中，我们找到了经典超几何级数满足的某些乘积公式的有限域类似。我们把两个高斯超几何级数的乘积表示为与高斯超几何级数。我们利用高斯和雅可比和的性质以及有限域Appell级数的工作，推导出高斯超几何级数满足的乘积公式。然后，我们使用这些变换显式地评估一些特殊的值和高斯超几何级数。通过计算有限域上CM椭圆曲线上的点，Ono发现了包含平凡和二次特征作为参数的-and-Gaussian超几何级数的某些特殊值。后来，埃文斯和格林发现了某些高斯超几何级数的特殊值，其中包含任意字符作为参数，其中小野获得的一些值作为特殊情况。我们表明，埃文斯和格林的一些结果遵循我们的产品公式，包括一个有限域模拟的经典克劳森的身份。
In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two-Gaussian hypergeometric series as- and-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of- and-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of- and-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain-Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen’s identity.