Proof of some conjectural supercongruences of Guo and Schlosser
type: Item
instance of: scholarly article
title: Proof of some conjectural supercongruences of Guo and Schlosser
zbMATH Open document ID: 1470.11005
author: Arijit Jana
author: Gautam Kalita
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: This paper deduces some some supercongruence relations concerning trancated hypergeometric series, and in particular, establishes some supercongruences conjectured by Guo and Schlosser. More precisely, the following supercongruence relation is proved for truncated hypergeometric series $_5F_4$. Theorem 1. Let $p>3$ be a prime. Let $n$ be a positive integer and $x$ a non-zero integer such that $p\equiv x\pmod n$ with $|x|<n<p$. If $\frac{x}{n}+\frac{1}{2}$ is not non-positive integer, then \[_5F_4\left(\begin{matrix} \frac{x}{n}& \frac{x+2n}{2n}& \frac{1}{2}&\frac{x}{n}&\frac{x}{n}\\ && \frac{x}{2n}&\frac{x}{n}+\frac{1}{2}&1&1\end{matrix}\,|\, 1\right) \equiv (-1)^{\frac{p-x}{n}}+\frac{p+1}{2}\frac{p}{x}\Gamma_p\left(\frac{1}{2}-\frac{x}{n}\right) \Gamma_p\left(\frac{1}{2} +\frac{x}{n}\right) \pmod{p^3}.\] Here $\Gamma_p$ is the $p$-adic Gamma function. There are several corollaries to Theorem 1. Corollary. 1. Let $p\equiv 1\pmod 3$. Then \[\sum_{k=0}^{(p-1)/3} (6k+1)\frac{(\frac{1}{3})_k^4(2k)!}{k!^4(\frac{2}{3})_{2k}}\equiv p\pmod{p^3}.\] 2. Let $p\equiv 2\pmod 3$. Then \[\sum_{k=0}^{(p+1)/2} (6k-1)\frac{(-\frac{1}{3})^4_k(2k)!}{k!^4(-\frac{2}{3})_{2k}}\equiv p\pmod{p^3}.\] Theorem 2. Let $p\geq 5$ be a prime. Then \[\sum_{k=0}^{(p-1)/2} (-1)^k(4k+1)\frac{(\frac{1}{2})_k(-\frac{1}{2})_k^2}{(k+1)^2 k!} \equiv \begin{cases} 0\pmod{p^2}\quad\mbox{if }p\equiv 3\pmod 4\\ \frac{8p}{3}\pmod{p^2}\quad\mbox{if }p\equiv 1\pmod 4.\end{cases}\] Corollaries and Theorem 2 establish the conjectures posed by \textit{V. J. W. Guo} and \textit{M. J. Schlosser} [``Some $q$-supercongruences from transformation formulas for basic hypergeometric series'', Preprint, \url{arxiv:1812.06324}]. Proofs use some hypergeometric series identities.
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 33C20
zbMATH DE Number: 7344950
zbMATH Keywords: hypergeometric series
zbMATH Keywords: supercongruences
zbMATH Keywords: gamma function
reviewed by: Noriko Yui
MaRDI profile type: Publication
full work available at URL: s11139-019-00221-5
OpenAlex ID: W3013952348
cites work: Q4222737
cites work: Supercongruences for truncated hypergeometric series and p-adic gamma function
cites work: Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties
cites work: Number Theory
cites work: Q4040884
cites work: \(q\)-analogues of two ``divergent'' Ramanujan-type supercongruences
cites work: A \(q\)-analogue of a Ramanujan-type supercongruence involving central binomial coefficients
cites work: Some new \(q\)-congruences for truncated basic hypergeometric series
cites work: A \(q\)-microscope for supercongruences
cites work: Some congruences on truncated hypergeometric series
cites work: Congruences concerning truncated hypergeometric series
cites work: On some conjectures of Swisher
cites work: Supercongruences for sums involving rising factorial
cites work: On some supercongruences concerning truncated hypergeometric series
cites work: On Van Hamme's (A.2) and (H.2) supercongruences
cites work: Semi-automated proof of supercongruences on partial sums of hypergeometric series
cites work: Hypergeometric evaluation identities and supercongruences
cites work: Some supercongruences occurring in truncated hypergeometric series
cites work: Supercongruences occurred to rigid hypergeometric type Calabi-Yau threefolds
cites work: A $p$-adic supercongruence conjecture of van Hamme
cites work: Q5508372
cites work: On the supercongruence conjectures of van Hamme
cites work: Q4357441
cites work: Some supercongruences involving
cites work: Ramanujan-type supercongruences
DOI: 10.1007/S11139-019-00221-5
description: scientific article; zbMATH DE number 7344950