Some \(q\)-supercongruences modulo the fifth power of a cyclotomic polynomial from squares of \(q\)-hypergeometric series
type: Item
instance of: scholarly article
title: Some \(q\)-supercongruences modulo the fifth power of a cyclotomic polynomial from squares of \(q\)-hypergeometric series
zbMATH Open document ID: 1490.11006
author: Hanfei Song
author: C. Wang
published in: Results in Mathematics
publication date: 2021-11-24T00:00:00Z
review text: Soon after having introduced the ``creative microscoping'' method together with \textit {W. Zudilin} [Adv. Math. 346, 329--358 (2019; Zbl 1464.11028)], \textit {V. J. W. Guo} applied it along with the Chinese remainder theorem for coprime polynomials in order to find \(q\)-supercongruences modulo the \textit {fourth} power of a cyclotomic polynomial [Adv. Appl. Math. 120, Article ID 102078, 16 p. (2020; Zbl 1456.11024)]. Following the same pattern, but considering the truncated sums of the squares of \(q\)-hypergeometric series like \textit{M. El Bachraoui} [Ramanujan J., 54, 415--426 (2021; Zbl 1473.11002)], here the authors focus on \(q\)-supercongruences modulo the \textit {fifth} power of a cyclotomic polynomial. Specifically, they establish six new \(q\)-supercongruences modulo \([n]\Phi_n(q)^4\) and they conjecture the extension modulo \([n]^2\Phi_n(q)^3\) for four of them, where \([n]=[n]_q=(1-q^n)/(1-q)=1+q+\cdots+q^{n-1}\) is the \(q\)-integer of any positive odd integer \(n\) and \(\Phi_n(q)=\prod_{j=1}^n (q-\zeta^j)\) (being \(n\) and \(j\) relatively prime, with \(\zeta=e^{2\pi i/n}\) a \(n\)-th root of unity) is the \(n\)-th cyclotomic polynomial in \(\mathbb{Z}q\). Beyond four relations supplied by \textit {C. Wei} [``Some \(q\)-congruences modulo the fifth and sixth powers of a cyclotomic polynomial'', Preprint, \url{arXiv:2104.07025}], a theorem from \textit{V. J. W. Guo} and \textit{W. Zudilin} [J. Math. Anal. Appl., 475, 1636--1646 (2019; Zbl 1445.11014)], a technique by \textit{V. J. W. Guo} and \textit{M. J. Schlosser} [Constr. Approx. 53, No. 1, 155--200 (2021; Zbl 1462.33006)], a lemma from \textit{H.-X. Ni} and \textit{L.-Y. Wang} [Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., RACSAM 116, No. 1, Paper 30 (2022; Zbl 1480.11003)], and de L'Hôpital's rule, the proof employs the \(q\)-Pfaff-Saalschütz summation formula and the Watson's \({}_8\phi _7\) transformation both available in the \textit {Encyclopedia of Mathematics and its Appli…
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 7432907
zbMATH Keywords: \(q\)-supercongruence
zbMATH Keywords: creative microscoping method
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: Chinese remainder theorem
zbMATH Keywords: \(q\)-hypergeometric series
zbMATH Keywords: truncated sum
MaRDI profile type: Publication
full work available at URL: s00025-021-01536-w
OpenAlex ID: W3213417290
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DOI: 10.1007/S00025-021-01536-W
description: scientific article; zbMATH DE number 7432907