A \(q\)-supercongruence from a \(q\)-analogue of Whipple's \({}_3F_2\) summation formula
type: Item
instance of: scholarly article
title: A \(q\)-supercongruence from a \(q\)-analogue of Whipple's \({}_3F_2\) summation formula
zbMATH Open document ID: 1506.11007
author: Chuanan Wei
published in: Journal of Combinatorial Theory. Series A
publication date: 2022-11-29T00:00:00Z
review text: By means of the Chinese remainder theorem for coprime polynomials, the L'Hôpital rule, the \(q\)-analogue of two summation formulas for basic hypergeometric series (Chu-Vandermonde and Whipple's \({ }_{3} F_{2}\)), two lemmas (one 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 the other from the author [J. Comb. Theory Ser. A 182, Article ID 105469, 15 p. (2021; Zbl 1468.11009)]), and the `creative microscoping' method introduced by \textit {V. J. W. Guo} and \textit {W. Zudilin} [Adv. Math. 346, 329--358 (2019; Zbl 1464.11028)], the paper proves the following supercongruence modulo \([n] \Phi_{n}(q)^{4}\), \[ \begin{multlined} \sum_{k=0}^{M}(-1)^{k}[4k+1] \frac{\left(q ; q^{2}\right)_{k}^{4}\left(q^{2} ; q^{4}\right)_{k}}{\left(q^{2} ; q^{2}\right)_{k}^{4}\left(q^{4} ; q^{4}\right)_{k}} q^{k} \\ \equiv \begin{cases}{[n] \frac{\left(q^{2} ; q^{4}\right)_{(n-1) / 4}^{2}}{\left(q^{4} ; q^{4}\right)_{(n-1) / 4}^{2}}\left\{1+[n]^{2} \sum_{j=1}^{(n-1) / 2} \frac{(-1)^{j+1} q^{2 j-n}}{[2 j]^{2}}\right\},} & \text { if } n \equiv 1 \pmod 4, \\ {[n]^{2} q^{(1-n) / 2} \frac{\left(q^{3} ; q^{4}\right)_{(n-1) / 2}}{\left(q^{5} ; q^{4}\right)_{(n-1) / 2}},} & \text { if } n \equiv 3 \pmod 4,\end{cases} \end{multlined} \] where \(n\) is a positive integer, \(\Phi_{n}(q)\) is the \(n\)-th cyclotomic polynomial in \(q\), and \(M=(n-1) / 2\) or \(n-1\). As corollary, the author establishes modulo \(p^{s+4}\) that \[ \begin{multlined} \sum_{k=0}^{m}(-1)^{k}(4 k+1) \frac{(1 / 2)_{k}^{5}}{k !^{5}} \\ \equiv \begin{cases}\frac{(1 / 2)_{\left(p^{s}-1\right) / 4}^{2}}{(1)_{\left(p^{s}-1\right) / 4}^{2}}\left\{p^{s}+\frac{p^{3 s}}{4} H_{\left(p^{s}-1\right) / 2}^{(2)}-\frac{p^{3 s}}{8} H_{\left(p^{s}-1\right) / 4}^{(2)}\right\}, & \text { if } p^{s} \equiv 1 \pmod 4, \\ p^{2 s} \frac{(3 / 4)_{\left(p^{s}-1\right) / 2}}{(5 / 4)_{\left(p^{s}-1\right) / 2}}, & \text { if } p^{s} \eq…
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 33D15
zbMATH DE Number: 7624751
zbMATH Keywords: basic hypergeometric series
zbMATH Keywords: \(q\)-supercongruences
zbMATH Keywords: cyclotomic polynomials
zbMATH Keywords: summation formulas
zbMATH Keywords: Chinese remainder theorem
zbMATH Keywords: creative microscoping method
zbMATH Keywords: harmonic numbers
zbMATH Keywords: Bernoulli numbers
MaRDI profile type: Publication
full work available at URL: j.jcta.2022.105705
OpenAlex ID: W4307488654
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DOI: 10.1016/J.JCTA.2022.105705
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description: scientific article; zbMATH DE number 7624751