A \(q\)-supercongruence from a \(q\)-analogue of Whipple's \({}_3F_2\) summation formula
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 \[ \begin{multlined} \sum_{k=0}^{m}(-1)^{k}(4 k+1) \frac{(1 / 2)_{k}^{5}}{k !^{5}} \\ \equiv \begin{cases}\frac{(1 / 2)_{\left(p^{s}-1\right) / 4}^{2}}{(1)_{\left(p^{s}-1\right) / 4}^{2}}\left\{p^{s}+\frac{p^{3 s}}{4} H_{\left(p^{s}-1\right) / 2}^{(2)}-\frac{p^{3 s}}{8} H_{\left(p^{s}-1\right) / 4}^{(2)}\right\}, & \text { if } p^{s} \equiv 1 \pmod 4, \\ p^{2 s} \frac{(3 / 4)_{\left(p^{s}-1\right) / 2}}{(5 / 4)_{\left(p^{s}-1\right) / 2}}, & \text { if } p^{s} \equiv 3 \pmod 4,\end{cases} \end{multlined} \] where \(p\) is an odd prime, \(s\) is positive integer, \(H_{m}^{(\ell)}\) is the harmonic numbers of \(\ell\)-order, and \(m=\left(p^{s}-1\right) / 2\) or \(p^{s}-1\). As auxiliary result, the author establishes a connection between this paper and the works of \textit {H. Swisher} [Res. Math. Sci. 2, Paper No. 18, 21 p. (2015; Zbl 1337.33005)] and \textit {J.-C. Liu} [J. Math. Anal. Appl. 471, No. 1--2, 613--622 (2019; Zbl 1423.11015)] through several properties of the \(p\)-adic Gamma function \(\Gamma_{p}(x)\), some of which provided by \textit {H. Pan}, \textit {R. Tauraso} and \textit {C. Wang} [J. Reine Angew. Math. 790, 53--83 (2022; Zbl 1501.11057)] and by \textit {C. Wang} and \textit {H. Pan} [Math. Z. 300, No. 1, 161--177 (2022; Zbl 1498.11070)], and via a threefold contribution about the harmonic numbers, in relation to Bernoulli numbers, supplied by \textit {Z.-H. Sun} [Discrete Appl. Math. 105, No. 1--3, 193--223 (2000; Zbl 0990.11008)], [J. Number Theory 128, No. 2, 280--312 (2008; Zbl 1154.11010)].
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