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Some congruences involving fourth powers of central q-binomial coefficients

Part of: Combinatorics Sequences and sets

Published online by Cambridge University Press:  30 January 2019

Victor J. W. Guo  and
Su-Dan Wang
Victor J. W. Guo
Affiliation:
School of Mathematical Sciences, Huaiyin Normal University, Huai'an, Jiangsu223300, People's Republic of China (jwguo@hytc.edu.cn)
Su-Dan Wang
Affiliation:
Department of Mathematics, East China Normal University, Shanghai200062, People's Republic of China (sudan199219@126.com)
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Abstract

We prove some congruences on sums involving fourth powers of central q-binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]:

$$\sum\limits_{k = 0}^{((p^r-1)/(2))} {\displaystyle{{4k + 1} \over {{256}^k}}} \left( \matrix{2k \cr k} \right)^4\equiv p^r\quad \left( {\bmod p^{r + 3}} \right),$$
where p⩾5 is a prime and r is a positive integer. Our method is similar to but a little different from the WZ method used by Zudilin to prove Ramanujan-type supercongruences.

Keywords

q-binomial coefficientsq-WZ methodcyclotomic polynomialsq-analogue of Wolstenholme's binomial congruenceq-analogue of Morley's congruence

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions 05A30: $q$-calculus and related topics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1127 - 1138
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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