q-Analogues of Two Supercongruences of Z.-W. Sun

被引：15

作者：

Gu, Cheng-Yang ^{[1
]}

Guo, Victor J. W. ^{[1
]}

机构：

[1] Huaiyin Normal Univ, Sch Math & Stat, Huaian 223300, Jiangsu, Peoples R China

来源：

CZECHOSLOVAK MATHEMATICAL JOURNAL | 2020年 / 70卷 / 03期

基金：

中国国家自然科学基金;

关键词：

congruences; q-binomial coefficient; cyclotomic polynomial; Franklin's involution; FINITE GENERALIZATIONS; NUMBERS;

D O I：

10.21136/CMJ.2020.0516-18

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We give several different q-analogues of the following two congruences of Z.-W. Sun: Sigma(prX-1)/2 k=0 1 8k 2k k ! 2 pr (mod p2) and (prX-1)/2 k=0 116k 2k k ! 3 pr (mod p2), where p is an odd prime, r is a positive integer, and ( mn) is the Jacobi symbol. The proofs of them require the use of some curious q-series identities, two of which are related to Franklin's involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.

引用

页码：757 / 765

页数：9

共 50 条

[21] Proof of a Conjecture of Z.-W. Sun on Trigonometric Series
Sun, Brian Y.
Meng, J. X.
[J]. UTILITAS MATHEMATICA, 2017, 102 : 223 - 229
[22] Another Proof of a Supercongruence Conjecture of Z.-W. Sun
Gautam Kalita
[J]. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2021, 91 : 283 - 287
[23] On two double series for π and their q-analogues
Wei, Chuanan
[J]. RAMANUJAN JOURNAL, 2023, 60 (03): : 615 - 625
[24] q-Analogues of the (E.2) and (F.2) supercongruences of Van Hamme
Victor J. W. Guo
[J]. The Ramanujan Journal, 2019, 49 : 531 - 544
[25] Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum
Guo, Victor J. W.
Liu, Ji-Cai
[J]. JOURNAL OF NUMBER THEORY, 2015, 156 : 154 - 160
[26] q-Analogues of the (E.2) and (F.2) supercongruences of Van Hamme
Guo, Victor J. W.
[J]. RAMANUJAN JOURNAL, 2019, 49 (03): : 531 - 544
[27] PROOF OF A CONGRUENCE FOR HARMONIC NUMBERS CONJECTURED BY Z.-W. SUN
Mestrovic, Romeo
[J]. INTERNATIONAL JOURNAL OF NUMBER THEORY, 2012, 8 (04) : 1081 - 1085
[28] Some q-analogues of supercongruences for truncated 3F2 hypergeometric series
Guo, Victor J. W.
[J]. RAMANUJAN JOURNAL, 2022, 59 (01): : 131 - 142
[29] Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials
Guo, Victor J. W.
Zeng, Jiang
[J]. JOURNAL OF NUMBER THEORY, 2012, 132 (08) : 1731 - 1740
[30] On two kinds of q-analogues of generalized Stirling numbers
Xu, Aimin
Tang, Peipei
[J]. RAMANUJAN JOURNAL, 2017, 43 (02): : 371 - 381

← 1 2 3 4 5 →