The Rodriguez-Villegas type congruences for truncated q-hypergeometric functions

详细信息    查看全文

• 作者：Victor J.W. Guo^a ; ^{jwguo@hytc.edu.cn} ; Hao Pan^b ; ^{haopan79@zoho.com} ; Yong Zhang^c ; ^{yongzhang1982@163.com}
• 关键词：primary ; 11B65 ; secondary ; 05A10 ; 05A30
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：358-368
• 全文大小：279 K
• 卷排序：174

文摘

In this paper, we establish a Rodriguez–Villegas type congruence for truncated q-hypergeometric functions. Using this result, we can confirm several conjectures of Guo and Zeng, such as∑k=0p−1(q;q3)k(q2;q3)k(q3;q3)k2≡(−3p)q1−p23(mod(1+q+⋯+qp−1)2), where p⩾5p⩾5 is a prime, (a;q)n=(1−a)(1−aq)⋯(1−aqn−1)(a;q)n=(1−a)(1−aq)⋯(1−aqn−1), and (⋅p) denotes the Legendre symbol modulo p.