A new \(q\)-extension of the (H.2) congruence of Van Hamme for primes \(p\equiv 1\pmod{4}\)
This work enriches the impressive list of papers inspired by the famous congruence (H.2) from \textit {L. van Hamme} [Lect. Notes Pure Appl. Math. 192, 223--236 (1997; Zbl 0895.11051)] \[ \sum^{(p-1)/2}_{k=0} \frac{\left( \frac{1}{2}\right)^3_k}{k!^3} \equiv \begin{cases} -\Gamma_p \left( \frac{1}{4}\right)^4 \quad \pmod{p^2} & \text{if } p\equiv 1 \pmod{4},\\ 0 \pmod{p^2} & \text{if } p\equiv 3 \pmod{4}, \end{cases} \] where \(p\) is an odd prime, \((x)_k=x(x+1) \cdots (x+k-1)\) represents the Pochhammer symbol, and \(\Gamma_p (x)\) is the \(p\)-adic Gamma function. More specifically, the author is motivated by the refinement of \textit {L. Long} and \textit {R. Ramakrishna} [Adv. Math. 290, 773--808 (2016; Zbl 1336.33018)] \[ \sum^{(p-1)/2}_{k=0} \frac{\left( \frac{1}{2}\right)^3_k}{k!^3} \equiv \begin{cases} -\Gamma_p \left( \frac{1}{4}\right)^4 \quad \pmod{p^3} & \text{if } p\equiv 1 \pmod{4},\\ -\frac{p^2}{16} \Gamma_p \left( \frac{1}{4}\right)^4 \quad \pmod{p^3} & \text{if } p\equiv 3 \pmod{4}, \end{cases} \] whose \(q\)-extension of the case \(p\equiv 1 \pmod{4}\) is the main purpose of this paper. Recalling a \(q\)-congruence found by \textit{V. J. W. Guo} [Result. Math. 76, No. 2, Paper No. 109, 12 p. (2021; Zbl 1470.11034)], the author establishes that, for \(n\equiv 1 \pmod{4}\) and \(n>1\), \[ \sum_{k=0}^{(n-1)/2} \frac{(q;q^2)_k^2 (q^2;q^4)_k}{(q^2;q^2)_k^2 (q^4;q^4)_k}q^{2k} \equiv [n] \frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}+[n]^3 \sum_{k=0}^{(n-3)/2} \frac{(1+q^{2k+1}) (q^3;q^4)_k}{[2k+1]^2 (q^5;q^4)_k}q^{2k+1} \pmod{\Phi_n(q)^3}, \] where \([n]=(1-q^n)/(1-q)\) is the \(q\)-integer, \((x;q)_n\) is the \(q\)-Pochhammer symbol, and \(\Phi_n(q)\) is the \(n\)th cyclotomic polynomial in \(q\). The author remarks that such achievement differs from a similar result due to \textit{C. Wei} [Result. Math. 76, No. 2, Paper No. 92, 9 p. (2021; Zbl 1470.33015)]. The author also proves two auxiliary theorems by employing the substitution \(q^n=1+t(q) \Phi_n(q)\), an identity about the \(n\)th harmonic numbers \(H_n^{(m)}\) of order \(m\) he computed via \texttt{SIGMA}, and additional congruences involving \(H_n^{(m)}\) supplied by himself in collaboration with \textit {H. Pan} [Math. Z. 300, No. 1, 161--177 (2022; Zbl 1498.11070)].
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