Some \(q\)-supercongruences modulo the fifth power of a cyclotomic polynomial from squares of \(q\)-hypergeometric series
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 Applications}; namely, in the Appendices of Volume No. 96 [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. This paper is closely connected to a work by \textit{V. J. W. Guo} and \textit{L. Li} [``\(q\)-Supercongruences from squares of basic hypergeometric series'', Preprint, \url{arXiv:2112.12076}] whose main results are here recalled, confirmed and refined.
scientific article; zbMATH DE number 7432907
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