论文标题
高斯高几何序列的某些产品公式和值
Certain product formulas and values of Gaussian hypergeometric series
论文作者
论文摘要
在本文中，我们发现经典高几幅系列满足的某些产品公式的有限场类似物。我们将两个$ {_ 2} f_1 $ -gaussian超几何系列的产品表示为$ {_ 4} f_3 $ - 和$ {_ 3} f_2 $ -gaussian超单几何系列。我们使用高斯和雅各比总和的属性，以及我们在有限田Appell系列上的早期作品来推断高斯超几何系列满足的这些产品公式。然后，我们使用这些转换来显式评估$ {_ 4} f_3 $ - 和$ {_ 3} f_2 $ -gaussian超几何系列的一些特殊值。通过计数有限字段的CM椭圆曲线上的计数点，Ono找到了$ {_ 2} f_1 $的某些特殊值 - 和$ {_ 3} f_2 $ -gaussian超测量序列，包含琐碎和二次字符作为参数。后来，埃文斯（Evans）和格林（Greene）发现了某些$ {_ 3} f_2 $ -gaussian超几何序列的特殊值，其中包含任意字符作为参数，从ONO获得的某些值遵循特殊情况。我们表明，埃文斯（Evans）和格林（Greene）的某些结果来自我们的产品公式，包括经典克劳森（Clausen）身份的有限场类似物。
In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two ${_2}F_1$-Gaussian hypergeometric series as ${_4}F_3$- and ${_3}F_2$-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 ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of ${_2}F_1$- and ${_3}F_2$-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain ${_3}F_2$-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.
